Uh~x7 < £-GL<> (^aU^sf- THE, THEORY OF SUBSTITUTIONS AND ITS APPLICATIONS TO ALGEBRA. By DR. EUGEN NETTO, Professor of Mathematics in the University of Giessen. Revised ton l.he Author and Translated with his Permission By F. N. COLE. Ph. D., Assistant Professor of Mathematics in the \ University of Michigan. ANN ARBOR, MICH.: THE register publishing company. TIbe llnlanb press. 1802. QNIVERSITY OP ^IJFORNJA PREFACE. The presentation of the Theory of Substitutions here given differs in several essential features from that which has heretofore been custom- ary. It will accordingly be proper in this place to state in brief the guiding principles adopted in the present work. It is unquestionable that the sphere of application of an Algorithm is extended by eliminating from its fundamental principles and its general structure all matters and suppositions not absolutely essential to its nature, and that through the general character of the objects with which it deals, the possibility of its employment in the most varied directions is secured. That the theory of the construction of groups admits of such a treatment is a guarantee for its far-reaching impor- tance and for its future. If, on the other hand, it is a question of the application of an aux- iliary method to a definitely prescribed and limited problem, the elab- oration of the method will also have to take into account only this one purpose. The exclusion of all superfluous elements and the increased usefulness of the method is a sufficient compensation for the lacking, but not defective, generality. A greater efficiency is attained in a smaller sphere of action. The following treatment is calculated solely to introduce in an elementary manner an important auxiliary method for algebraic inves- tigations. By the employment of integral functions from the outset, it is not only possible to give to the Theory of Substitutions, this operat- ing with operations, a concrete and readily comprehended foundation, but also in many cases to simplify the demonstrations, to give the various conceptions which arise a precise form, to define sharply the principal question, and — what does not appear to be least important — to limit the extent of the work. The two comprehensive treatises on the Theory of Substitutions which have thus far appeared are those of J. A. Serret and of C. Jordan. The fourth section of the "Algebre Superieure " of Serret is devoted to this subject. The radical difference of the methods involved here and there hardly permitted an employment of this highly deserving work for our purposes. Otherwise with the more extensive work of Jordan, the "Traite" des substitutions et des Equations algebriques." Not only the new fundamental ideas were taken from this book, but it is proper to mention expressly here that many of its proofs and pro^ IV PREFACE. >es of thought also permitted of being satisfactorily employed in the- present work in spite of the essential difference of the general treat- ment. The investigations of Jordan not contained in the "Traits" which have been consulted are cited in the appropriate places. But while many single particulars are traceable to this "Traits" and to these investigations, nevertheless, the author is indebted to his honored teacher, L. Kronecker, for the ideas which lie at the foundation of his entire work. He has striven to employ to best advantage the benefit which he has derived from the lectures and from the study of the works of this scholarly man, and from the inspiring personal inter- course with him; and he hopes that traces of this influence may appear in many places in his work. One thing he regrets: that the recent im- portant publication of Kronecker, "Grundzuge einer arithmetischen Theorie der algebraischen Grossen," appeared too late for him to derive from it the benefit which he would have washed for himself and his readers. The plan of the present book is as follows: In the first part the leading principles of the theory of substitutions are deduced with constant regard to the theory of the integral func- tions; the analytical treatment retires almost wholly to the background, being employed only at a late stage in reference to the groups of solvable equations. In the second part, after the establishment of a few fundamental principles, the equations of the second, third and fourth degrees, the Abelian and the Galois equations are discussed as examples. After this follows a chapter devoted to an arithmetical discussion the necessity of which is there explained. Finally the more general, but still elementary questions with regard to solvable equations are examined. Stkassburg, 1880. To the preceding I have now to add that the present translation differs from the German edition in many important particulars. Many new investigations have been added. Others, formerly included, which have shown themselves to be of inferior importance, have been omitted. Entire chapters have been rearranged and demonstrations simplified. In short, the whole material which has accumulated in the course of time since the first appearance of the book is now turned to account. In conclusion the author desires to express his warmest thanks to Mr. F. N. Cole who has disinterestedly assumed the task of translation and performed it with care and skill. EUGEN NETTO. GipsSEN, 1892. TRANSLATOR'S NOTE. The translator has confined himself almost exclusively to the function of rendering the German into respectable English. My thanks are especially due to The Register Publishing Company for their gener- ous assumption of the expense of publication and to Mr. C.N. Jones, of Milwaukee, for valuable assistance while the book was passing through the press. F. N. COLE. Ann Arbor, February 27, 1892. TABLE OF CONTENTS. PART I. Theory of Substitutions and of Integral Functions. CHAPTER I. SYMMETRIC OR SINGLE-VALUED FUNCTIONS ALTERNATING AND TWO-VAL- UED FUNCTIONS. / 1-3 . Symmetric and single-valued functions. 4 . Elementary symmetric functions. 5-10. Treatment of the symmetric functions. 11. Discriminants. 12. Euler's formula. 13. Two- valued functions; substitutions. 14. Decomposition of substitutions into transpositions. 15. Alternating functions. 16-20. Treatment and group of the two-valued functions. CHAPTER II. MULTIPLE-VALUED FUNCTIONS AND GROUPS OF SUBSTITUTIONS. / % 22 . Notation for substitutions. 24 . Their number. 25 . Their applications to functions. 26-27. Products of substitutions. 28. Groups of substitutions. 29-32 . Correlation of function and group. 34. Symmetric group. 35. Alternating group. 36-38. Construction of simple groups. 39-40 . Group of order pf. CHAPTER III. THE DIFFERENT VALUES OF A MULTIPLE-VALUED FUNCTION AND THEIR ALGEBRAIC RELATION TO ONE ANOTHER. ty *{ 41-44 . Relation of the order of a group to the number of values of the corresponding function. Ylll CONTENTS. U>. Croups belonging to the different values of a function. 16 17. Transformation. 48-50. The Cauchy-Sylow Theorem. 51. Distribution of the elements in the cycles of a group. 52. Substitutions which belong to all values of a function. 53. Equation for a ^-valued function. 55. Discriminants of the functions of a group. 50-59. Multiple-valued functions, powers of which are single- valued. CHAPTER IV. TRANSITIVITY AND PRIMITIVITY. SIMPLE AND COMPOUND GROUPS. ISOMORPHISM. 7 * 60-61. Simple transitivity. 62-63. Multiple transitivity. 64. Primitivity and non-primitivity. 65-67 . Non-primitive groups. 68. Transitive properties of groups. 69-71. Commutative substitutions; self-conjugate subgroups. 72-73. Isomorphism. 74-76. Substitutions which affect all the elements. 77-80. Limits of transitivity. 81-85. Transitivity of primitive groups. 86. Quotient groups. 87. Series of composition. 88-89. Constant character of the factors of composition. 91 . Construction of compound groups. 92. The alternating group is simple. 93 . Groups of order p\ 94. Principal series of composition. 95. The factors of composition equal prime numbers. 96. Isomorphism. ( .»7 98. The degree and order equal. 99-101. Construction of isomorphic groups. CHAPTER V. ALGEBRAIC RELATIONS BETWEEN FUNCTIONS BELONGING TO THE SAME GROUP. — FAMILIES OF MULTIPLE-VALUED FUNCTIONS. / / *f 103-105. Functions belonging to the same group can be rationally expressed one in terms of another. 106. Families; conjugate families. 107. Subordinate families. CONTENTS. IX 108-109. Expression of the principal functions in terms of the subordinate. 110. The resulting equation binomial. 111. Functions of the family with non-vanishing discriminant. CHAPTER VI. THE NUMBER OF THE VALUES OF INTEGRAL FUNCTIONS. /*< 8 112. Special cases. 113. Change in the form of the question. 114-115 . Functions whose number of values is less than their degree. 116. Intransitive and non-primitive groups. 117-121. Groups with substitutions of four elements. 122-127 . General theorem of C. Jordan. CHAPTER VII. CERTAIN SPECIAL CLASSES OF GROUPS. / Lf. U 128 . Preliminary theorem. 129 . Groups ft with r — n — p. Cyclical groups. 130. Groups £2 with r = n=ji- q- 131. Groups ft with r = n-p 2 . 132-135. Groups which leave, at the most, one element unchanged.— Metacyclic and semi-metacyclic groups. 136. Linear fractional substitutions. Group of the modular equations. \ 137-139. Groups of commutative substitutions. CHAPTER VIII. ANALYTICAL REPRESENTATION OF SUBSTITUTIONS. THE LINEAR GROUP. / (y O 140. The analytical representation. 141 . Condition for the defining function. 143. Arithmetic substitutions. 144. Geometric substitutions. 145. Condition among the constants of a geometric substitution. 146-147. Order of the linear group. PART II. Application of the Theory of Substitutions to the Algebraic Equations. CHAPTER IX. the equations of the second, third and fourth degrees. GROUP OF AN EQUATION. RESOLVENTS. ' ° ° 148. The equations of the second degree. X CONTENTS. 1 19. The equations of the third degree. The equations of the fourth degree. The general problem formulated. Galois resolvents. 154. Aifect equations. Group of an equation. 156 Fundamental theorems on the group of an equation. Group of the Galois resolvent equation. 158 159. eneral resolvents. CHAPTEK X. THE CYCLOTOMIC EQUATIONS. ■ 161 . I definition and irreducibility. 162. Solution of cyclic equations. 163. Investigation of the operations involved. 164-165 Special resolvents. 166. < onstruction of regular polygons by ruler and compass. The regular pentagon. 168. The regular heptadecagon. 169 170. Decomposition of the cyclic polynomial. CHAPTEK XL THE ABELIAN EQUATIONS. / f 7 171-172. One root of adequation a rational function of another. 173. Construction of a resolvent. 171-17.";. Solution of the simplest Abelian equations. 17C Employment of special resolvents for the solution. 177. Second method of solution. 180. Examples. 181 . Abelian equations. Their solvability. 182. Their group. 183. Solution of the Abelian equations; first method. 184 186. Second method. 187. Analytical representation of the groups of primitive Abelian equations. 188-189. Examples. CHAPTER XII. EQUATIONS WITH RATIONAL RELATIONS BETWEEN THREE ROOTS. U, 2. -2, 190-193. Groups analogous to the Abelian groups. 194. Equations all the roots of which are rational functions of t wo among them. 196. Their group in the case n —p. I'.i7 . The binomial equations. CONTENTS. XI 199. Triad equations. 200-201 . Constructions of compound triad equations. 202. Croup of the triad equation for n = 7. 203-205. Group of the triad equation for n = 9 206 . Hessian equation of the ninth degree. CHAPTER XIII. THE ALGEBKAIC SOLUTION OF EQUATIONS. 207-209. Rational domain. Algebraic functions. 210-211. Preliminary theorem. 212-216. Roots of solvable equations. 217. Impossibility of the solution of general equations of higher degrees. 218. Representation of the roots of a solvable equation. 219. The equation which is satisfied by any algebraic expression. 220-221. Changes of the roots of unity which occur in the expres- sions for the roots. 222-224 . Solvable equations of prime degree. CHAPTER XIY. THE GROUP OF AN ALGEBRAIC EQUATION. 226. Definition of the group. 227. Its transitivity. 228. Its primitivity. 229. Galois resolvents of general and special equations. 230. Composition of the group. 231 . Resolvents. 232-234. Reduction of the solution of a compound equation. 235. Decomposition of the equation into rational factors. 236-238. Adjunction of the roots of a second equation. CHAPTER XV. ALGEBRAICALLY SOLVABLE EQUATIONS. 239-241 . Criteria for solvability. 242 . Applications. 243. Abel's theorem on the decomposition of solvable equations. 244. Equations of degree p k ; their group. 2 ir ; . Solvable equations of degree p. 246. Solvable equations of degree p 2 . 248-249. Expression of all the roots in terms of a certain number of them. JiHo 166 ±%L ERRATA. p. 7, footnote, for transformatione read transmutatione. p. 15, line 10, read

. p. 29, line 12, for '/'read . p. 29, line 9, from bottom, for read W. p. 31, line 8, read G = [1, {x x x 2 ) (x 3 x^), (x^) (x 2 x t ), (x^) (x 2 x 3 )~\. p. 41, line 7, for p read p f . p. 52, line 13, read a

T . _ p. 52, line 5, from bottom, for — read -4~. d a d a p. 89, line 2, for not more read less. p. 93, line 9, for a group H read a primitive group H. p. 94, line 2, for n — q -\- 2 read n — q-\-k. p. 98, line 19, for %J>' b $ c read § a 3Vc p. 101, line 3, for il, read £>', . p. 103, line 14, read [1, (z,2 2 )]- p. 125, Theorem XI, read: In order that there may be a pp-valued function / a prime power y p of which shall have p values, etc. p. 159, lines 10, 11 from bottom, read: Since ,...,. the s of which belongs to 1\ , must be prime to r x . p. 166, line 4, read r a , t 2 r 2 , t 3 r 2 , . . . p. 174, line 2, for c, read 2cj. p. 210, foot note, for No. XI, etc., read 478-507, edition of Sylow and Lie. p. 219, line 10, for t (cos a) read x (cos a). p. 224, line 2 from bottom, read: which leaves two elements with successive indices unchanged. p. 248, line 3 from bottom, read V** + Y = F a + l (V a + a , . . .). PART I. THEORY OF SUBSTITUTIONS AND OF THE INTEGRAL FUNCTIONS. CHAPTER I. SYMMETRIC OR SINGLE -VALUED FUNCTIONS. ALTERNA- TING AND TWO-VALUED FUNCTIONS. § 1. In the present investigations we have to deal with n ele- ments x u x 2 , . . . x n , which are to be regarded throughout as entirely independent quantities, unless the contrary is expressly stated. It is easy to construct integral functions of these elements which are unchanged in form when the x^B are permuted or interchanged in any way. For example the following functions are of this kind : 3Jj — (— .To -p Xj -p . . . ~~p X n , ■•\ a x -f + ^i a x£ + • • • + x ?~ &f + x i a x f + • • • I '**/< X l - • • I <&n ■' ,; — l j ' '' -''j)" \. X l ^3) '-''j X i)~ ' • • \ X 'i — \ X n)r etc. Such functions are called symmetric functions. "We confine ourselves, unless otherwise noted, to the case of integral functions. If the ,r A *s be put equal to any arbitrary quantities, a } , a 2 , . . . a, , so that a\ = a ]: x 2 = a,, . . . x„ = a„, it is clear that the symmetric functions of the x^s will be unchanged not only inform, but also in value by any change in the order of assignment of the values a a. to the X\B. Such a reassignment may be denoted bv x 1 = a h , x 2 = «,_,, . . . x„ = a, n where the a ([ , a,,, . . . denote the same quantities a„ a 2 , ... in any one of the possible n ! orders. Conversely, it can be shown that every integral function, . = o all the coefficients a A are equal to zero, then /(•<') vanishes identi- cally, i. e., f(.v) is equal to zero for every value of x. Conversely, if /*(.') vanishes for every value of x, then all the coefficients d\ are equal to zero. For if f(x) is not identically zero, then there is a value '~„ such that for every real x of which the absolute value x is greater than ~ , the value of the function /(a?) is different from zero. For £ we may take the highest of the absolute values of the several roots of the equation /(.*■) = 0. Without assuming the existence of roots of algebraic equations, we may also obtain a value of "„ as follows:* Let a k . be the numerically greatest of the n coefficients a , o t , a, , in (1), and denote - ' by r. We have then '"" ''' "•" '""" ( x\" '+.<" -'+... + 1, a, < o k — a„ < a k = <-',, 1 a" I — 1 < I — 1 Hence, for any value of x not lying between - r and + r, a ,-'•" ' +a„ ,r 2 + .. +a -i i ., - < so that the sign of /(.<•) is the same as that of a x". Consequently, we may take T„ = /•. • i. Kr :cker. i irelle 101, i>. :ut. SYMMETRIC AXD TWO-VALUED FUNCTIONS. ■> (B). If no two of the integral functions (2) /i (*),/.(*),.../-(*) are identically equal to each other, then there is always a quantity ~„ such that for every x the absolute value of which is greater than £ , the values of the functions (2) are different from each other. For, if we denote by r a3 the value determined for the function fa (•«') — //3 (#), as r was determined in (A), we may take for ~„ the greatest of the quantities r a p. (C). If in the integral function /(■'•,. ■''_., . . . ■<•„) "S«.«. . . * r.M r.M \i A all the coefficients a are equal to zero, then the function/ vanishes identically, *. e., the value of / is equal to zero for every system of values of x t , x 2i . . . x„. To prove the converse proposition we put (3) •<•,, = g, x, = gv, x, = g*, . . . x„ = gv n ~ l .i\ xl x 2 Ki . . . .»„ A "- then becomes a power of g, the exponent of which is J-A! A, • • • Km = /, + K V + V + • • • KS"- 1 From (B), we can find a value for v such that for all greater val- ues of v, the various ?' Ai a: • • • Km are all different from one another. "We have then /(,*-!, a-,, . . . x m ) = 2 a MX , . . . Km g r * *«■•.• *« But, from (A), if all the coefficients a do not disappear, we can take g so large that / is different from 0. The converse proposition is then proved. (D). If a product of integral functions (4) /, (as n ar«, . . . #„) fi{x u x 2 , . . . x n ) . . . f m {x u x 2 , . . . x n ) is equal to zero for all systems of values of the £c A 's, then one of the factors is identically zero. For, if we employ again the substitution (3), we can, from (C), select such values g a and v a for any factor f a (a?j , x 2 , ... x n ) which does not vanish identically, that for every system of values which arises from (3 ) when g > g a and v > > a the value of f a is different from zero. If then we take g greater than g u g 2 , . . . g„ and at the same time v greater than v t , v 2 , ... v„ we obtain systems of values 4 THEORY OF StTBSTITUTIONS. of the .c A 's for which (4) does not vanish, unless one of the factors vanishes identically. The proof of Theorem I follows now directly from (C). For if c .»,,.... x i is a single-valued function, and if e-, (.<•,. .r, . . . a?„) arises from 95 by any rearrangement of the .r A 's, then it follows from tin* fact that c has only one value, that the difference C (•<",, .t\ , . . . X„ ) CT] ( Xj , • <'_>, . . . X n ) vanishes identically. If the elements x K are not independent, Theorem I is no longer necessarily true. For instance, if all the .r A 's are equal, then any arbitrary function of the X\S is single-valued. Again the function r ; j.iy' -<'j — 4a?] .tv + 3 •<'/ is single-valued if a?, — 2.r.,, although it is unsymmetric. § 3. If, in any symmetric function, we combine all terms which only differ in their coefficients into a single term, and consider any one of these terms, Cxfx^xJ, ..., then the symmetric character of the function requires that it should contain every term which can be produced from the one considered by any rearrangement of the • "-. If these terms do not exhaust all those present in the func- tion there will still be some term, C'xf' x.f .c/ ... in which the sys- tem of exponents is not the same as in the preceding case. This term then gives rise to a new series of terms, and so on. Every symmetric function is therefore reducible to a sum of simpler sym- metric functions in each of which all the terms proceed from any single one among them by rearrangement of the X\S. The several terms of any one of these simple functions are said to be of the same type or similar. Since these functions are deducible from a single term, it will suffice to write this one term preceded by an S. Thus S (./ ) denotes in the case of two elements .»',- -f- .<•_.-', in the case of three . [nation of the n th degree, this equation, apart from a constant fac- tor, has the form (5) f(x) = (x .'•,)(..• .<•,)...(.<• — x n ) = the left member of which expanded becomes SYMMETRIC AND TWO-VALUED FUNCTIONS. O (6) .<■ (.r, +£c a + . .. +-<•„) a?"- 1 1 v'*I '*'-' I '''l '' ; ~~ *^2 «**3 I • • • I '''«-I &,■< ■•• which does not vanish is positive. This amounts then to assigning an arbi- trary standard order of precedence to the elements .r A . In accordance with this convention, c l5 c 2 , c 3 , . ... C\ } . . . have for their highest terms respectively it 1 ■ tASt (< 1 * *K> 1 kK. i (A/Q • • • ■ tA 1 it i M/Q • ■ « i,i \ ^ • * • and the function of cf c 3 ? . . . has for its highest term faa + ? + y + . . . x ^ + y + . . . ^y + . . . In order, therefore, that the highest terms of the two expressions, Cj a cf c :1 v . . . and c, a ' cf c/ . . . may be equal, we must have « + ,5 + /-+... =«'+/3' + r'+... /J + r +...= ,5'+/+ ... r + • • • = /-'+•.. that is, a = «', ,5 = /?', j' = /, . . . It follows that two different systems of exponents in c x a cf cJ . . . give two different highest terms in the X\8. Again it is clear that xfxfxj ... {a>fi> r >d m . m ) is the highest term of the expression cf^ cf~ vc 3 y~ & . . . and that * Demonstratio nova altera etc. Gesammelte Werke III. § 5, pp. 37-38. Cf. Kron- ecker. Monatsberichte der Berliner Akademie, 1889, p. 943 seq. 6 THEORI OF SUBSTITUTIONS. all the terms in the expansion of this expression in terms of the .r A "s are of the same degree. § 6. If now a symmetric function S be given of which the highest term is A xfxfxfxf ... (a ' .' ,'"'-...) the difference S — A cf-tcf— * c s v~ s . . . = N, will again be a symmetric function ; and if, in the subtrahend on the left, the values of the c A 's given in (7) be substituted, the highest term of S will be removed, and accordingly a reduction will have been effected. If the highest term of S, is now A 1 ,x\ a ' xf .'•;/' .<•/' . . . , then s } — a\ c, a ' ~ v cf - y c.y -»...= s 2 is again a symmetric function with a still lower highest term. The degrees of S 2 and S t are clearly not greater than that of S, and since there is only a finite number of expressions xfxf .<■■/ ... of a given degree which are lower than xf -v.f .ry . . ., we shall finally arrive by repetition of the same process at the symmetric function ; that is S k —A k c/ k) -^ k) c/ k) -y {k) ... = 0; and accordingly we have s = a, cf-ficf-y . .. -\-A 2 c l af -i i 'c 2 f , '-y'... + . . . § 7. It is also readily shown that the expression of a symmetric function of the .r A 's as a rational function of the C\8 can be effected in only one way. For, if an integral symmetric function of « , . .« c, could be reduced to two essentially different functions of c n <•_, , ... c„, V ''i, c 2 , . . . c„) and (c,, <-,,. . . c„), then we should have, for all values of the .r A 's, the equation V ('-,, '•_., . . . c„) = c'-fC,,?,. ... c n ) The difference n ) B) s,. — c 1 s r _ 1 +c 2 s,._ 2 — ... + (— l) r rc r =0 (r^w) These two formulas can be proved in a variety of ways. The formula A) is obtained by multiplying the right member of (6) by x r ~ n , replacing a? by x k , and taking the sum over / = 1, 2, . . .n. The formula B ) may be verified with equal ease as follows. If we represent the elementary symmetric functions of x 2 , r, , ....-• by c/, c 2 , . . . c'„_], we have C \ — &1 ~T~ C l 1 @2 = '- '*'l C \ T" C 2 5 C 3 -— fy C l T C 8) and accordingly, if r < n, we have ,■', * = 1, 2, . . . n - - 1; ;>. = 2, 3, . . . n) Au = ( .r, .r_,j ( .r, #"3,)" (.^1 - -*'i I • • • ( '''i ■''„ ) (8) ('.<•, — •'-,; i J I X., .r 4 r. . . i.e. .«•„)-' ( ■''„ _ 1 3C„) • This quantity -J already satisfies the condition as to the equality of the X\8, and, being the simplest function with this property, is itself the discriminant. It contains .' n (n - 1) factors of the form ' — .r^'f; its degree is )i (n — 1), and the highest power to which any X\ occurs is the (n -l) th . It is the square of an integral, but, as we shall presently show, unsymmetric function, with which we shall hereafter frequently have to deal. £ 12. Finally we will consider another symmetric function in which the discriminant occurs as a factor. Let the equation of which the roots are x u x 2 , . . . x„ be, as before, f(x) = 0. Then if we write SYMMETRIC AND TWO- VALUED FUNCTIONS. 11 we have, for all values X = 1, 2, . . . n, the equation f'(x K ) = U' A — .»',) (x x — a-,) . . . (j-a — .r.v_ ,) (.r A - -,)... (■'\ — .<•„)• We attempt now to express the integral symmetric function Slxf ./'(*,) ./'(.-• ,)... /'(a--)] in terms of the coefficients c,, Co, . . . c„ of / (.r), Every one of the n terms of S is divisible by .r, — x 2 , since either / '(.»-,) or/V- | occurs in every term. Consequently, by the same reasoning as in § 11, 5 is divisible by (x x — a? 2 ) 2 , and therefore being a symmetric function, by every (.».■„ — Xp) 2 , that is by -J =TIVa - » K ) a ( A < " ; ^ = 1, 2, . . . n — 1 ; ac = 2, 3, . . . »). Am S is therefore divisible by the discriminant of f{ '.»■), /. e., by the dis- criminant of the n roots of / (.<•). Now/(.r A ) is of degree of n — 1 in X\ and of degree 1 in every other x^\ and therefore x i a • /'O-') • /'( v z) ■ ■ • /'(•*'..) is of degree a + n- — 1 in .<-, ■''/' • f'( x i) ■ f'i x z)' • • f'( x n) is of degree 2n — 3 in .r, . Consequently, if a < n — 1, ,$' is of degree 2 n — 3 in ,i\ , while J is of degree 2 n — 2 in a\. But since J is a divisor of S, it fol- lows that S is in this case identically 0. (9) S[.•„)-'. In J the coefficient of ay" ~" 2 is (.»■_. .<■)-' (.»■_, — x t y. . . (>„_, — x r . —j > The desired numerical factor is therefore ( — 1) ^~ and we have 12 THEORY OF SUBSTITUTIONS. (10) s [..■,•■-'. /'(■•■■)•/'(■'■,). ../'(*«)] = (- Formulas (9) and (10) evidently still hold if we replace a?i° or a?, " ~ ' by any integral function ,» ^''.> i **i "i •' • The first two functions are unchanged if .r, and x 3 be inter- changed, the second also if .»', and x 5 be interchanged, etc. Functions are designated as one- two-, three-, in-valued according to the number of different values they take under the operation of all the n ! possible substitutions. The existence of one- valued functions was apparent at the outset. "We enquire now as to the possibility of the existence of two-valued functions. In § 11 we have met with the symmetric function J, the dis- criminant of the n quantities .*,, x 2 , . ■ ■ ■''„■ The square root of J is also a rational integral function of these n quantities: *The formula D is due to Euler; C;ilc. Inf. IL g L169. SYMMETRIC AND TWO-VALUED FUNCTIONS. IS I X 2 ■'' , ) I ■''_• ■''» ) • • • ' •' '.' ■' .. ' I C ■'', )...(•<';- X I Every difference of two elements x a — .''p occurs once and only once on the right side of this equation. Accordingly if we inter- chancre the X\8 in any way, every such difference still occurs once and only once, and the only possible change is that in one or more cases an x a — .(' 3 may become Xp — x a . The result of any substitution is therefore either + s/ J or — \ / J > *• , •''., • . . •''„ into the order ,r (l , .r,„, a?f 8 , . . . x-, n , we apply first the transposition (.c, . , ). The order of the .r A 's then becomes .<•,-,, .<•_, , .r :; , . . . x^ — i .<•, . .<•/, + 1 , ... x„ , and we have now only to convert the order .-■_, . . . .»',, _ b x u X(j + i, ... x„ into the order ..-,•_.. xi s , . . . .<',„. By 14 THEORY OF SUBSTITUTIONS. repeating the same process as before, this can be gradually effected, and the theorem is proved. Since a symmetric function is unaltered by any substitution, we obtain as a direct result Theorem V. A function which is unchanged by every transposition is symmetric. ij 15. There is therefore at least one transposition which changes the value of any alternating function into the opposite value. We will denote this transposition by ( x a Xp) , and the alter- nating function by 4', and accordingly we have {X X ,X 2 ,. . . .<•„.. . . 07/s,. . . X„)= —tp (.r,, ..■_,, . . . Xp, . . . X a - i Accordingly, if .*' a = Xp , we must have 4> = 0. Consequently the equation 4> (x 1} x 2 , ... Z, ... Xp, ... x„) = regarded as an equation in z has a root z = Xp and the polynomial 4> is therefore divisible by z — Xp . The function (p (.r,, ,»•,,, . . . x a . . . Xp . . . ■>■„) therefore contains x a - - Xp as a factor, and, consequently, c' " con- tains (x a — Xpf as a factor. But since, for all substitutions,

is either a one- or a two-valued function, since every substitution SYMMETRIC AND TWO-VALUED FUNCTIONS. 15 either leaves both numerator and denominator unchanged or changes the sign of one or both of them. But this quotient cannot be two- valued, for then it would be again divisible by s/ J , which is con- trary to hypothesis. It must therefore be symmetric, and we have accordingly Now r if m were an even number, the right member of this equa- tion, and consequently the left, w r ould be symmetric. We must therefore have m = 2n -f- 1. And if we write S l . -I" = S. we have -f ^ = S . a/1 Corollary. From the form of an alternating function it folloics that such a function remains unchanged or is changed in sign simultaneously with s/ A for all substitutions. § 16. Having now shown how to form all the alternating func- tions, we proceed to the examination of the two-valued functions in general. Let cr (•<',, aj 2 j . . . x„) be any two-valued function, and let the tw T o values of

osili<»is change the value %/ J into — s/ A ; all substitutions which are formed from an even number of transpo- sitions leave \/ A unchanged. Similar results hold for all two- val- ued functions. SYMMETRIC AND TWO-VALUED FUNCTION 3. 17 £ L9. Every substitution can be reduced to a series of transpo- sitions in a great variety of ways, as is readily seen, and as will be shown in detail in the following Chapter. But from the preceding theorem it follows that the number of transpositions into which a substitution is resolvable is always even, or always odd, according as the substitution leaves \/ J unchanged or changes its sign. Theorem IX. If a given substitution reduces in one way to an even (odd) number of transpositions, it reduces in every way to an even (odd) number of transpositions. §20. Theorem X. Every two-valued function is the root of an equation of the second degree of which the coefficients are rational symmetric functions of the elements x u U' 2 , . . . x n . From the equations of § 17, 9l = St + S, V J, 9a = 'Si — S 3 V4 we have for the elementary symmetric functions of 2 CT 2 = S{ J S-{ . We recognize at once that ^ and c._> are the roots of the equa- tion r — 2 s, 9 + (s; 1 — j s.f) = o. It is however to be observed here that it is not conversely true that every quadratic equation with symmetric functions of the X\8 as coefficients has two-valued functions, in the present sense, as roots. It is further necessary that the roots should be rational in the elements X\ , and this is not in general the case. CHAPTER II. MULTIPLE-VALUED FUNCTIONS AND GROUPS OF SUBSTITUTIONS. § 21. The preliminary explanations of the preceding Chapter enable us to indicate now the course of our further investiea- tions, at least in their general outline. Exactly as we have treated one-valued and two-valued functions and have determined those substitutions which leave the latter class of functions unchanged, so we shall have further, either to establish the existence of functions having any prescribed number of values, or to demonstrate their impossibility; to study the algebraic form of these functions; to determine the complex of substitutions which leave a given multiple- valued function unchanged; and to ascertain the relations of the various values of these functions to one another. Further, we shall attempt to classify the multiple -valued functions; to exhibit them possibly, like the two-valued functions, as roots of equations with symmetric functions of the elements as coefficients; to discover the relations between functions which are unchanged by the same sub- stitutions; and so on. § 22. At the outset it is necessary to devise a concise notation for the expression of substitutions. Consider a rational integral function of the n independent quan- tities .»•,, .'•_,, . . . .«■„, which we will denote by

f'V^) = {.r,.v t ) (a- 5 X 6 ) (x,.r ; ) i.r..c-) (.,•,.-„) (.r_,.r 7 ) (.r,..-,) (.<•,.,•,) (.,■„.■ , Since the given substitution resolves into 3, 5, 9, ... , transposi- tions, always an odd number, we have here an example of the prin- CORRELATION OF FUNCTIONS AND GROUPS. 21 ciple of Chapter I, § 19, and it appears that this substitution changes the sign of \/ J. The third method gives us § 24. We determine now the number of all the possible substi- tutions by finding the entire number of possible permutations. Two elements .»', , x 2i can form two different permutations, .»-,.»• , and .)'„/', . If a third element x s be added to these two, it can be placed, 1) at the beginning of the permutations already present .c..i\.r,, .(■..(•,.*■,, or 2) in the middle: ■f l .r :i .r,, .''..'•..•<",, or 3) at the end: .r,.r_,.r,, .r .r,.c. . There are therefore 2-3 = 3! permutations of three elements. If a fourth element be added, it can occur in the first, second, third or fourth place of the 3 ! permutations already obtained, so that from every one of these proceed 4 new permuta- tions. There are therefore in this case 2-3-4 = 4! permutations, and again, for 5 elements, 5!, in general, for n elements, ;;! permu- tations. If now, in the notation A, we take for the upper line the natural order of the elements, .v,. .<•_., x .., . . . x„, and for the lower line suc- cessively all the n ! possible permutations, we obtain all the possible distinct substitutions of the n elements. It is to be noticed that among these there is contained that sub- stition for which the upper and lower lines are identical. This substitution does not affect any element; it is denoted by 1. and regarded asunity or as the identical substitution. Theorem I. For n elements there are n\ possible substitu- tions. To obtain the same result from the notation B more elaborate investigations would be necessary for which this is not the place: in case of the notation C it is easy to establish the number it ! by the aid of induction. AVe arrive in the latter case at a series of interesting relations, of which at least one may be noted here. If a substitution in the expression for which all the elements occur contains a cycles of a elements, b cycles of ,5 elements, 22 THEORY OF SUBSTITUTIONS. where aa + 6]J+ ... =n, N I we can obtain from this by rearrangement of the order of the cycles and by cyclical permutation of the elements of each single cycle a! a" \>\j h . . . expressions for the same substitution. Consequently there are n! a! a a b\ p . . . distinct substitutions which contain a cycles of a elements, b cycles of /3 elements, and so on. The summation of these numbers with respect to all possible modes, N), of distributing the number n gives us all the possible n ! substitutions. Hence '^ 1 _-. * £ 25. If now we apply all the n\ substitutions to the function

i\ expres- sions, including that produced by the substitution s, = 1. These expressions we may denote by 9^ = 9l, 9*,, 9s 3 ■ ■ ■ 9e a ■ ■ ■ f • or simply, where no confusion is likely to occur, by 9\, 9i, 9z, • • • 9a, • • • 9n- These values are not necessarily all different from one another. Some of them may coincide with the original value f ('.<•, , .r,, . . . x n ). We direct our attention at first to the complex of those substitu- tions which do not change the value of c. If

'., r„ and suppose *Cauchy: Exercices d'analyse, III. 17.;. CORRELATION OF FUNCTIONS AND GROUPS. 23 i This function is unchanged by 8 of the possible 24 substitutions, namely by S[ - — 1, S_. -- (XiSCqff S :i — (QCzXiJ) S t — (iT,X_>j (.('..*',), S- f — \ tA ]•' jU VCa ), Sq — I JC^JC^JC^JC^ I, o-j — I ■' [• By supposition there is no substitution s' different from s^s.^s,, ... s, , which leaves the value of

>s (?) s n s 2 , s 8 , . . . s r . The succession of the operations in a product •- = s a Sps y . . . is to be reckoned from left to right. § 27. The expression of such a product in the cycle notation which we have adopted is obtained as follows: If the two factors of a product are S a (•' ■•' ! ■'',,., . . . ) (•''/. .''/, •'' ...)..., sp = (x, x aiftj ...)!.'• .'■ •''...)..., then in s a Sp that element will follow x a by which Sp replaces a •„,. Suppose, for instance, that this element is x,, x . Again in s a Sp that element will follow x hi by which Sp replaces .*■„., . Let this be for example, x ki , etc. We obtain s S/3 = (cc a a? 7l j aVo • • ■ ) If the substitution s a be such that it replaces every index g of the elements a;, , x 2 , . . . x g , ... .»',,, by /„, and if sp be such that it replaces every index g by k, n or, in formulae, if sp = (x 1 x kl x khi . . . ) (x b x kb . . . ) . . .. then the product will be of the form S a Sp = faXj^ ...) I.r .r ...)... The following may serve as an example: 8 a ~ '■'' ■'';•'': .) '-''j-'';)} Sp - l.'\.<' ( .f, | (.''.'■). S a .s^j = (.r,.r ) I. '■,.(•-. '•,./',, ) (.*'.) (.»■..'', I I' '• I'.''. I. W'fhave introduced here the expression "product." The ques- tion now arises how far the fundamental rules of algebraic multipli- cation a • b — b • a, a • (b • c) - (a • 6) • c remain valid in this case. An examination of this matter will show that the former, the commutative law, in general fails, while the second, the associative law, is retained. In fact the multiplication of CORRELATION OF FUNCTIONS AND GROUPS. 25 S a = ( .<', ■'., •'•„, ...)••., Sp = (.*'! »*, 0? fct] ...)..., as performed above shows that it is only in the special case where, for every a , i k = k, a , that the order of the two factors s a and s$ is indifferent. This occurs, for example, as is a priori clear, if the expressions for s„ and s$ contain no common elements. We may therefore interchange the individual cycles of a substi- tution in any way, since these contain no common elements. In the notation I? of page 19, on the other hand, this is not allowable. Passing to the associative law, however, if 8 a =(x s X u ) . . . , 8p = (x s X ks ) . . . , S y = {xjC h ...)..., we have the following series of products, Sp S y = (X s X lhs ...)..., S a 8p = X 8 X k . a ...)•• • «« (Sj3 S y ) = (x s X lk . a ...)..., (s a Sf})Sy= {x s X lkia ...)..., from which follows Theorem II. In the multiplication of substitutions a col- lection of the factors into sab -products without change in the order of the factors, is permissible. An interchange of the factors, on the contrary, generally alters the result. Such an interchange is however permissible if the factors contain no common elements. § 28. From the preceding developments it appears that those substitutions Lr) S x = 1 , Sj , s 3 , . . . s a , . . . s,. which leave a given function

= (x v x'j) (•*■.;•''.)(•''-,■'',,), Sja r (.<•,. r..'\.r,) ( -'",•'■,, ), *!( — ( .*•,.<■,.<• ..f . ) ( ■''y.Etijj • s 'i:, = ('V'';) '''j-''|l '•'',■''.,), S 10 = yXiX t ) (.<\.r.) I.r.r, ). The order of the group of

4 ele- ments, the order of the corresponding group becomes 8 • (n - 1 ) ! , the group being obtained by multiplying the 8 substitutions of § 25 by all the substitutions of the elements .«■-, .«•„ , ... x , CORRELATION OF FUNCTIONS AND GROUPS. 27 § 29. The following theorem is obviously true : Theorem III. For every single- or multiple-valued func- tion there is a group of substitutions which, (implied to thefunction r leave it unchanged. To show the perfect correlation of the theory of multiple-valued functions and that of groups of substitutions we will demonstrate the converse theorem : Theorem IV. For every group of substitutions there are functions which arc unchanged by all the substitutions of the group and by no others. We begin by constructing a function

■., If now two substitutions s a =(av<\ v ...)... and s^=(«r. f a , A . x . ..)..., on being applied to cr, gave the same result, we should have =

= * as was asserted. As to the second condition, the substitutions *,r, *-. g 8 r, . . . s r r are all different from *,, s 2 , s :! , ... s,., and consequently the func- tions < T . > : . c .. . . . cr. T are all different from, the factors •-' . . ■ • • f,,. of 'I'. Moreover, this difference is such that no c- - can be equal to the product of a <- by a constant r, , in which case

is impracticable, since the multiplication soon becomes unmanageable even for moderately large values of r. There is however, another process of construc- tion in which the product is replaced by a sum, and every difficulty of calculation is removed. We begin by taking as the basis for further construction, instead of the linear function c, the following function V (.<•,,. r.,..- :! , . . . .r„) = x^xpxp . . . .<•„%. The ap's are to be regarded here, as before, as arbitrary quanti- ties, and, as the ,r A 's are also arbitrary, it follows at once that

n + H + K + ■■■ +K- The proof of the correlation of G and ip proceeds then exactly as in the preceding Section in the case of G and ( P. Remark. — By making certain assumptions with respect to the a's we can assign to the v'''s some new properties. Thus we may select the a's in such a way that an equation between any two arbi- trary systems of the a's, «,- x + "/, + ■ • • «; A = «*, + «/,, + • ■ • «^ , necessarily involves the equality of the separate terms on the right and left. This condition is satisfied if, for example, we take 30 THEORY OF SUBSTITUTIONS. a, > «, , « :i > a, + a., , « 4 > «, -(- a 2 -J- a, , , in particular if atj = 1, Wo = 2, « :j =4, a 4 = 8, «-, = 10, .... E. g. If a,-, -f- a,., -f- . . . -f «, A = 13, we must have i\ = 1, i, = 3, *3 = 4, X = 3. Example. — We will apply the two methods given above to the familiar group G = [l, Ov*',.), (a? a a? 4 ), (ayr,) (x^ 4 ), (.r,x 3 ) (a-,.'-, ,, (a ■,.«■» | (avB 8 ), (.!•,. JVC..*-.), (.»■,.»■,.»•...«•..)], ( n = 4, V = 8) taking as fundamental functions c — Xl -\- ix 2 — x 3 — ia\ and 4' — •>'\"'>\' ' where, as usual, i = V — 1. We have then the following results: <1> = (.r, + /..•, — x s — ix 4 ) (.r., + ix x — x 3 — ijc 4 ) (a 1 , + It., — .<•, ix 3 ) <•'".■ + '-''i — Xi—iXa) (ops + iXi — .*-,— *.»■,) (.1-, + /.(•; — x % — /.*-,) (x z -\- /.'•, — x 2 — ix x ) {Xi,-\-ix 3 — Xj — /.r_,) = [(.r, + ix z — x 3 — ix t ) (x 2 J t-ix 1 —x 3 —ix^ (x L -\-ix 2 — x A — ix 3 ) (•'•:+ KB] X t lXs)~f = { [(«!— xtf + ix,— x t )*] l{x l -x i ) 2 + {x 2 —x i f] }\ W = .,•_..,- .,■? + x 1 x 3 2 x i 3 + avr 4 V + a^V + a^V + ■'• ■<; x = (ajj + as,) (xfxf + xM + fa + xd («iV + »»V) = (a^ + .'-J (•'■ ! x^{x 3 x? + x?x 2 2 ). Neither of the two methods furnishes simple results directly. But from 'I' we may pass at once to the function [(.<-, — Xi f + (£c 8 — x 4 )-] [(«, — * 4 ) a + (x 2 — .!•:; )-] . and from '/'to the two functions (.!■, +.r, )(.(■. -)-.(-,) and .(■,.!■_. + . r,-',, the latter being already known to us. It is clear also that by alter- ing the exponents which occur in W we can obtain a series of func- tions all of which belong to G. Among these are included all functions of the form (.-y + x 2 " I (aV + <*-*) . a\ a «% a + •«•.;" ■''"■ CORRELATION OF FUNCTIONS AND GROUPS. 31 In general we perceive that to every group of substitutions there belong an infinite number of functions. It may be observed however that we cannot obtain all functions belonging to a given group by the present methods. Thus the function belongs to the group G = [1, (.'■,..-,), (.»■,.<•,!, (.<■,.,•_.) ('■;.'■,)], but cannot be obtained by these methods. More generally, if the functions c''', 0", c""', . . . belong respectively to the groups H', H", H'", . . . , and if the substitutions common to these groups (Cf. § 44, Theorem VII) form the given group G, then the function where the a's are arbitrary, belongs to the group G. § 32. We now proceed to consider the case where the elements x x , x 2 , . . . x„ are no longer independent quantities. Theorem V. Even where amj system of relations exists among the elements x x , x> , ... x„, excluding only the case of the equality of two or more elements, tee can still construct n\-valued functions of x n .('_,, . . . X„.* Using the notation of the preceding Section, we start from the same linear function iUv, — av,) + a.Jx^ — Xr.) . . . + a ,/- 4. A series of other anal- ogous results will also be obtained. For the present we shall concern ourselves only with the con- struction and the properties of some of the simplest, and for our purpose, most important groups. * § 34. First of all we have the group of order n ! , composed of all the substitutions. This group belongs to the symmetric func- tions, and is called the symmetric group. In Chapter I we have seen that every substitution is reducible to a series of transpositions. Accordingly, if a group contains all * Cf. Serrel : Cours d'algl bre Bup6rieure. II, ss 116-420. Oauchy: loc. cit. CORRELATION OF FUNCTIONS AND GROUPS. 33 the transpositions, it contains all the possible substitutions and is identical with the symmetric group. To secure this result it is how- ever sufficient that the group should contain all those transpositions which affect any one element, for example a?, , that is the transposi- tions (■<•,•'••>, (•'•,•'•;>, (XjX t ), . . . (•*•,.*•„). For every other transposition can be expressed as a combination of these n 1 ; in fact every (x a Xfi) is equivalent to a series of three of the system above, (where it is again to be noted that the order of the factors is not indifferent). We have then Theorem VI. A group of n elements x t , x 2 , ... x„ which contains the n — 1 transpositions (X a .X\), [XgXzJi . ■ . [X a X a — 1/) (^a^'a + 1 )) • • • ( '*a',i ) is identical with the symmetric group. Corollary. A group ivhich contains the transpositions (jt'aJCp), (x a X y ), . . . (x a X#) contains all the substitutions of the symmetric group of the elements § 35. We know further a group composed of all those substitu- tions which are equivalent to an even number of transpositions. For all these substitutions, and only these, leave every two-valued func- tion unchanged, and they therefore form a group. We will call this group the alternating group. Its order r is as yet unknown, and we proceed to determine it. Let I) s, = 1, s 2 , s 3 , ... s,. be all the substitutions of the alternating group, and let II) «/, s,', s,', ... s/ be all the substitutions which are not contained in I), and which are therefore composed of an odd number of transpositions. We select now any transposition t, for example . t, that is r = t. Again, since I) and II) contain all the substitutions, r-\-t = n\. Hence n\ 2 We will note here that there is no other group /' of order -j • For a function y>, belonging to such a group would be unchanged by ".,' substitutions, and would be changed by all others. It would therefore possess other values beside , into

v, V, ....^.i], then <:l<>>iqx to the two- valued functions. We can generalize this proposition. The proof, being exactly parallel to the preceding, may be omitted. CORRELATION OF FUNCTIONS AND GROUPS. 35 Theorem VIII. Either all, or exactly half of the substi- tutions of every group belong to the alternating group. Corollary. Those substitutions of any given group of order r which belong to the alternating group, form a group within the given group, the order of which is either r or f) . The simplest substitutions belonging to the alternating group contain three elements in a single cycle, (x a XpX y ). They are equiv- alent to two transpositions, (x a XpX y ) = (x a ^y) (xpX y ). A substitution containing only one cycle (x^, x ki ... x lit ) we shall call a circular substitution of order m. Theorem IX. If a group of n elements contains the n — 2 circular substitutions it is either the alternating or the symmetric group. For since (XaXpXy) — (XjXoXp) (XjXzXp) (x v v,x y ) Uvr.,.r a ) (x^Xa) {x x x.pc^), it follows that the given group contains every circular substitution of the third order. And again, since '■''l''\-'":() (XiX i X 3 ) = (.»',.)•,) (.l';'' 4 ), (X } X 3 X 2 ) {XyV^.,) -- (XiX 3 ) \X 2 X 4 ), it follows that all substitutions occur in the given group which are composed of two, and consequently of four, six, or any even num- ber of transpositions. The theorem is therefore proved. We add the following theorem : Theorem X. If a group contains all circular substitutions of order m-\-2, it will contain also all those of order n>, and con- sequently it will contain either the alternating or the symmetric (/roup, according as m is odd or even. For we have [■I'^C.j . . ..'',„•'',,•''/,) [XiX'2 . . . • '',„•''„•''/,) (■'',„•'',„ — l • • • ■'_••';.■' \X a f ( .1 [.) j ... •',„_! •' ,„)• Finally, we can now give the criterion for determining whether a given substitution, expressed in cycles, belongs to the alternating group, or not. The proof is hardly necessary. 36 HIEOHV or SUBSTITUTIONS. Theorem XT. If a substitution contains ni elements in /.• cycles, it does or does not belong to the alternating group according as in — k is even or odd. § 36. Any single substitution at once gives rise to a group, if we multiply it by itself /. e., if we form its successive powers. The meaning of the term "power" is already fully defined by the devel- opments of § 27. We must have s'" = s'"- 1 s = s ■ s'"- 1 = s m ~ 2 -s 2 — s a. . s fi . s >»-a-B — The process of calculation of the powers of a substitution is also clear from the preceding Sections. If wo wish to form the second, third, fourth, . . . « th power of a cycle, or of any substitution, we write after each element the second, third, fourth, . . . a th following element of the corresponding cycle, the first element of each cycle being regarded as following the last. Thus, from the cycle (■e,.c,.c,r v f- > . . . ) we obtain for the second power (.*•,.<■;,.)'- . . . ), for the third (.r,. !■,.<•; . . . ), for the fourth (.*',.<•-.(•,, . . . ), etc. It is obvious that in this process a cycle may break up into several. This will occur when and only when the number of elements of the cycle and the exponent of the power have a common divisor d > 1. The number of resulting cycles is then equal to d. For example, I .I'j. »'_,.('. .»',. *"-,-*",, ) - (•'V':-'\) I .»\. '',.!', ). ( ,l'|. I\. <'.,.<',. <',.'',,) ('» \'' / .' >• »• >• *• .' \ ,i i .« n* ..* i. ( p ,A |; f \ •* ]•* j^* -■* !•' {■' ■) J . If the mi miter of elements of a cycle be m, then the m"', (2m) tb , (3m) th , . . . poivers of the cycle, and no others, will be equal to 1. If a substitution contains several cycles with m l , m,, m 3 , . . . elements respectively, the lowest power of the substitution which is equal to 1 is that of which the exponent /• is the least common multiple of to, , m 2 , m,, . . . Thus | (.rV <• ) (i <•,./■,) l.r,,r,)\- = 1. This same exponent ;• is also the order of the group formed by the powers of the given substitution. For if we calculate CORRELATION OF FUNCTIONS AND GROUPS. 3 I S~, S 8 , . . . ST- 1 , 8' = 1, a furthor continuation of the series gives merely a repetition of the same terms in the same order: s'-+'=s, s'-+ 2 = s 2 , s r+3 = s 3 , . . . a 2 '- 1 = S r ~\ S* = 8 r =1, ... Moreover the powers of s from s 1 to s' are different from one another, for if s \ — s \+ m = s a . gl i (yt + /i<.r), then we should have contrary to hypothesis S* = 1 (/7. < r). The extension of the definition of a power to include the case of negative exponents is now easily accomplished. We write s~ k = s' _A ' = s'- v ~*'= . . . so that we have s k s~ k = 1. The substitution s A therefore cancels the effect of the substitu- tion s '', and vice versa. The negative powers of a substitution are formed in the same way as the positive powers, only that in forming ( — l) st , ( — 2) d , ( — 3) d , . . . powers, we pass backward in each cycle 1, 2, 3, . . . elements, the last element being regarded as next pre- ceding the first. It may be noted that (st)" } — t" 1 s~\ For (sf) _1 (st) =1 , and by multiplying the members of this equation first into / ~~ ' and then into s~\ we obtain the result stated. The simplest function belonging to the cycle (.r, .r, . . . x m ) is § 37. Given two substitutions s a and Sp , if we wish to deter- mine the group of lowest order which contains .s a and Sp, we have not only to form all the powers s a A , s^ and to multiply these together, but we must form all the combinations 1, s a \ V, s/V 1 ' -sp M s a A , s/s^s/, s^s/sp 1 ; . . . Of the substitutions thus formed we retain those which are dif- ferent from one another, and proceed with the construction until all substitutions which arise from a product of m factors are contained among the preceding ones. For then every product of m + 1 fac- :^S THEORY OF SUBSTITUTIONS. tors is obviously reducible to ono of m factors, and is consequently also contained among those already found. The group is then complete. /// case 8p8 a =a a Sp M j th e corresponding group is exhausted by all the substitutions of the form s a K Sp\ For we have in this case Sfi 8 a = SpS a Spf 1 = SfiS a - Sf — S a Sf, Sp m S a =8 a S^ i Sp m 'sa=Sa 2 Sp m ' l2 S a = S a 3 Sfr*\ S/3 S a —S a Sp * Consequently any product of three factors is reducible to a product of two. Thus S a P Sfi' T S a T =S a (P + T )S^ T % SpPSa'Sp^SSSfP+ti^, and the theorem is proved. For example, let ■ S 'l ~~ ~ (.l' r >'_,.('..' ',.'', ), .S'_, = ( • ' " ■ '•''-'' i ) j then ,S' ,.S'j - — { ,t j-^_i-' |**;j/ "i **J • The group of lowest order which contains s, and s 2 contains therefore at the most 5 • 4 = 20 substitutions. To determine whether the number is less than this, we examine whether it is possible thai 8 ] a S 2 P = 8{"8 2 S . If this were the case, it would follow that I Jut in the scries of powers of 8 2 there is only one which is also a pow- erof s n and this is the zero power. Consequently we must, have a pand/?= 8. The group therefore actually contains 20 substi tut ions. These are the following, where for the sake of simplicity we write only the indices: CORRELATION OF FUNCTIONS AND GROUPS. 39 Sl ° = 1, s 2 = (2351 ), .sv = (25) (34), a 3 3 (2 1 53 ), V = ( 1 2345), s,s 2 = (1325), rf = (15) (24), a,*. 3 = (1435), •V = (1 3524), Sl 2 s a = (1534), a,V - (14) (23), a, V = (1254), a, 8 = (14253), *,'*, = (1243), b,V = (13) (45), a^ 8 = (1523), .s,'^ (15432), a 1 *a 2 = (1452), Sj V = (12) (35), a^ 8 = (1342). Analogous results may be obtained, for example, for the case — ■ f y / y* / y* / y* / v f Y* / v* i c — ( y* v / y* "y* / y* , y* i I — V '1* ' ■ '< \ 5 6 tJi — V ' i 3*^ T"' 5*^- ti 7* /// case everj/ s^Sa (m=1, 2, 3, . . .) can be reduced to the form 8 a K Sp\ the group of lowest order which contains s a and Sp is exhausted by the substitutions of the form s a K 8p\ For by processes similar to those above we can bring every sub- stitution sp?s a v to the form s a K ,s j3 A . The proof is then reduced to the preceding. Furthermore if sp' is the lowest power in the series Sp, sp, . . . which occurs among the power of s a , then the group contains q times as many substitutions as the order A; of s a . For in the first place, if the exponent / in s a K Sp K is greater than q — 1, we can replace Sp* by a Sa^S/s", where v<.q — 1. There are therefore at the most q ■ k differ- ent substitutions s a K Sp\ Again if then we must have, if we suppose A > v, 8(3*-" = a.**-" (A— v which is a divisor of the product n\ = 1 • 2 • 3 . ., />, then there is a group of degree n and of order p. In the first place suppose n < p~, so that n = ap -\- h (a, b < p). Then, of the numbers 1, 2, 3, . . . n, only p, 2/», 3p, . . . are divisible by p, so that/= a. We select now from the n elements a systems of p betters each, aud form from each system a cycle, as follows: Then the group which arises from these is the group required : JS-\ = |^Sj , S, , ... .S._, ,S 2 , . . . S„ , S„ , . . . J = } Sj , So ■ . . . «S'„ j . For every s A with its various powers forms a subgroup of order p, and since no two of these a subgroups have any element in common, it follows from Theorem II that Sk< , s i ° = sSs k p, Accordingly every possible combination of substitutions s, a , s 2 ^, . . . belonging to K can be brought to the form s^s/s.y .... s a v (a, p, r , . . . v = 0, 1, 2, . . . p— 1). The group K therefore contains at the most p" substitutions. And it actually contains this number, for all these p" operations are dif- ferent from one another. For if sfsfsj . . . s a v = s^'s./'s-y . . . s/', it would follow that o — a'o a — o a — a' — .. 8'.. y> a v> ,. — Va — /oc. Q — y<, — (3 — a fi' ~ Po y' — y and therefore, since Sj , has no element in common with s a , S 3 , . . . , we must have a = «', etc. Again, if n=p 2 , we shall have/ = p + l, since in the series *In the designation of a group the brace, | | , as .distinguished from the bracket, T J< indicates that the group referred to is the smallest group which contains the included substitutions. The bracket contains nil the substitutions of the group considered, while the brace contains only the generating substitutions. The latter can generally be se- lected in many ways. df. the notation at the close of the last Section. 3a 42 THEORY OF SUBSTITUTIONS. 1, 2, 3, . . . p : , the numbers p, 2p, Sp, . . . (p — l)p, p* are divisible by p. We form now again the substitutions ,S', , s 2j .s, ( , . . . s I>} as before, and in addition to these the substitution .s,, ( . , which affects all the jr elements — I 'V 'v 'V* *■ t* »* ^y *>* 'V* o^ \ i> + i — V 'i *i 'i • • • * K i\ ,K >v *s • • • '.' *8 ' * * H ' ' ' ' i< )• Then the required group is K., = j «Sj , s 2 , . . . Sjj , s p _,_,(. For in the first place we can readily show that s l S j> + 1 — s i> + l s 2i ,S 'i' s V + l ~ S i' + l S a+u S p Sp + i -— Sp + , .S, , SA o O O « t* A C» Q O A V A q O O A 1 «*p + 1 — "> + 1*2 > *o *^ + 1 — Oj) + 1 *a + 1 5 *y< a p + 1 — d /> + 1 d l » SA r> - o - o A V A t! O * © A «*! " ij O - o A 1 S J» + 1 ' J> + 1 *3 » * a /> + 1 ./' + 1 a + 2 5 *{' B J> + 1 °i> + 1 ' - > 8j S,, + j* 1 = S^, + j** S^ + i , S a Sp + i = *p + ] s a + n > ^> s ^) + l "~" S ^ + 1 ,S y ' Accordingly every combination of the substitutions s t , s 2 , . . . 8 P +J can be brought, as in § 37, to the form s.^AV • • • V Sp + f (a, /9, r, . . . i, x = 0, 1, 2, . . . p— 1). But we must also show in the present case that we need only take the powers of s p + , as far as the ( p — l) Ul . We find that • S ',. ;-l — ~ '•''']•'''■_.•'''':; • • • "•**',,) {.^'k^ka ' • • Wb„) • • • Kp^ti^tn • • • **''«/ = S 1 8 2 . . . Sp, Sal' — r. a t . a o a p +1 — °l °i! • • • °p • Consequently, if k~>p, we can replace the highest power of *,, + i " which occurs in 8 p+1 k by powers of s,, s 2 , . . . s y ,, and these can then be written in the order above. The question then remains whether the p 1 ' + l =p f substitutions thus obtained are all distinct. If two of them wero equal .. a.. /5 o io < — a a 'v /3' o i'o k' .">! »2 • • • °jp °p + l — "l "a • • • °P -J'+l J we should have t . K — «' — a a' — a v fi' —B o i' — i * v , _j_] — *j »2 • • • *i» But the substitution on the right does not affect the first subscripts i, fc, . . . , while that on the left does, unless /. = /.'. The proof then proceeds as before. If n>p- but < i?, that is, if u = a/r -\- bp -{- c (a,b,c 3 +p, . . . (a— l)p t +p i , (a_l) p s+p, . . . (p — l)2r+p-=p\ so that in this case the multiplication of the p partial groups K, is not sufficient. In this case, exactly as for n = p\ we add another substitution which contains all the p ■ p' elements in a single cycle, and the p th power of which breaks up into the p substitutions as inr the case of s p + l above. Then, as in that case, we can show that the new group satisfies all the requirements. At the same time it is clear that the method here followed is perfectly general, and accord- ingly the theorem at the beginning of the Section is proved. § 40. Since all the groups K x , K 2 , K 3 , . . . enter into the forma- tion of the group iv, we have the following Corollary. // p J is the highest power of p which is a divi- sor of n\ , then ire can construct a series of groups of n elements 1, .K\ , K 2 , ... K\ , K\ + j , ... Ky which are of order respectively l,p,p\...p\p K + \ ...p\ "» Every group /\' A is contained as a subgroup in the next following K\ + 1 • CHAPTEE III. THE DIFFERENT VALUES OF A MULTIPLE-VALUED FUNC- TION AND THEIR ALGEBRAIC RELATION TO ONE ANOTHER. § 41. We have shown in the preceding Chapter, that to every function of n elements .r,, .r.,, x 3 v„ there belongs a group of substitutions, and that conversely to every group of substitutions there correspond an infinite number of functions of the elements. The examination of the relations between different functions which belong to the same group we reserve for a later Chapter. The problem which we have first to consider is the determination of the connection between the several values of a multiple-valued function and the algebraic relations of these values to one another. If

o so that rrr., leaves the function c, unchanged; consequently r lines of the table exhaust all the possible substitutions we deduce the following theorems : Theorem II. The order r of agroup (I of n elements is a divisor of n\ Theorem TTI. The number p of the values of an integral function of n elements is a divisor of n\ Theorem IV. The product of the number p of the values of an integral function by the order r of the corresponding group is equal to n ! The third theorem imposes a considerable limitation on the pos- sible number of values of a multiple valued function. Thus, for example, there can be no seven- or nine-valued functions of five elements. But the limits thus obtained are still far too great, as the investigations of Chapter V will show. § 43. Precisely the same method as that of § 41 can be applied to the more general case where all the substitutions of the group G belonging to tp are contained in a group H belonging to another function 4'i so that G is a part or subgroup of 11, just as in the special case above G was a subgroup of the entire or symmetric group. We see at once that all the substitutions of H can be arranged in a series of lines, each line containing r substitutions of the form * A ^, (; L, 2, . . . r). And we pass directly from the pre- ceding to the present case by reading everywhere for "all possible substitutions" simply "all the r, substitutions of H". We have then MULTIPLE -VALUED FUNCTIONS— ALGEBRAIC RELATIONS. 47 Theorem V. If all the r substitutions of the gran/, a are contained among those of a group H of order r, , then r is a divisor Theorem VI. Given hvo functions

. For we have _n\ ii\ >> r, r ' r { ' p x r ' Corollary. //' a function , and >.,, as was asserted. Again it is easily seen that all the substitutions of the second line are different from each other. For if a, ' s a ff 2 — ff.r ' Sfi ff P _ ' $»*»» » V ' Vp » • • • °f ' *^p] = ">~~ ' &1 V § 40. The functions c5 x , e,, . . . cr p are of precisely the same form and only differ in the order of arrangement of the X\S which enter into them. Such functions we have called (§3) similar or of the same type. Accordingly the corresponding groups G,, 2 , ... must also be similar or of the same type, that is they produce the same system of rearrangement of the elements .x\ and only differ in the order in which the elements are numbered. This is clear a priori, but we can also prove it from the manner of derivation of <*V~ "v? from s a , and in fact we can show that not only the groups Cr, , G 2 , . . . , but also the individual substitutions s a and trr~ 1 s a T ; are similar; that is, these two substitutions have the same number of cycles, each containing the same number of elements. The process of deriving the substitutions y ' . and finally .r, a by •<',, . Accordingly t, '.s a <7,- contains the cycle (a ' . . . .<\ a ), and this is obtained from the corresponding cycle (a?, •'•_ . . . 'a) of « a by regarding this cycle, so to speak, as a function of the elements .r A and applying to it the substitution a { . In the same way every cycle of J .{ ., .1 j . C' ;; — Xj •'', ~T~ •''.' •''; < respectively. By transposition with respect to ff a aU( l ff 8i we obtain from (.!-,. '•,■''■'•; )J, the two groups belonging respectively to c, and c- . . 1 1, (.«'|. <',:), i '■'',). (•'',•'' • (■'■■•'', ), (.r,.x 2 ) (. '■;.'■, ). ( .'■,.'', ) ( .'■ '■ ), [x 1 x 2 XgX i ), (.'■ - i 1 1 , [_1, (&1&4)) (•tV^i)? V^l**"-!/ (^V*^/* ( a V':;l '•'V'lh (•'',•''') '•'' -''i '• (.'V'::-''r'j'- £ 17. Corollarj T. // a group of substitutions is trans- formed with rrsjtcrt to (mi/ substitution whatever, the transformed siibsfifiit/n,,* form a gr-onj). Corollary II. The two, generally different, substitutions 8 a 8fi and * 3 .s a are similar. Fors a 80=8p l (s/sS a )s^.. MULTIPLE-VALUED FUNCTIONS ALGEBRAIC RELATIONS. 51 Corollary III. The substitution s a sps a ~ l is conjugate to sp ivith respect to s a ~\ Corollary IV. If the substitution s a is of order r and if sp be such that its 7"' \ + b c' a . let dc^-^b^r be any remaining value. Then in the same class with this belongs also &■ a , the order r of •Formulas A) and IS) were obtained by G. Frobeuius, Crelle 01. p. 281. as an extension of a result given by the Author, Math. Annalen XII!. MULTIPLE -VALUED FUNCTION'S — ALGEBRAIC RELATION^. ~> : ' the group is a multiple of p, and if a group contains a subgroup of order }> a where p is a prime number, the order of the group is a mul- tiple of p a . By the aid of the results of the preceding Section we can now also prove the converse proposition: Theorem IX. If p a be the highest power of the prime num- ber p which is a divisor of the order h of a group H, then H con- tain* subgroups of order p a . * In the demonstration we take for the G of the preceding Sec- tion the symmetric group, so that g = n\ . For H x we take the pres- ent group H, and for K t the group of order p J of § 39, Chapter II, p' being the highest power of p which is a divisor of n ! . The formula B) of § 48 then becomes The left member of this equation is no longer divisible byjp; con- sequently there must be at least one f$ = — which is also not di- dp visible by p ; that is dp is divisible by p a , and therefore H and Kp, the latter being a conjugate of K, have exactly p a substitutions in common, j" These form the required subgroup of H. Corollary. At the same time it appears that the group K con- tains among its subgroups every type of groups of order pi. For ire need only take any group of order pv for H iii tlie above demon- stration. § 50. The last theorem admits of the following extension : Theorem X. If the order h of a group H is divisible by pP, tlien H contains subgroups of order p&. The proof follows at once from Theorem XI, as soon as we have proved Theorem XI. Every group H of order p - contains a sub- group of order p a_1 . The corollary of the preceding Section permits us to limit the •Oauchy, loc. cit.. proved this theorem for the case a = 1. The extension to the case of any a was given by L. Sylow, Math. Annalen V. pp. .">S4-59L tFor every subgroup of K, and consequently every subgroup of K$. has for its order a power of p. 5 \ THEORY OF SUBSTITUTIONS. proof to the case of groups of order p a which occur as subgroups in the group of § 39, Chapter II. The group K = K f there obtained was constructed by the aid of a series of subgroups (§ 40) 1, ISTj , K 21 . . . K\ , K\ +1 , . . . K,_ , , of orders 1, p\ p\ ••• p\ P A+1 , ■•• P f ~\ every one of which is contained in the following one. If the group H occurs in this series, the theorem is already proved; if not, then let .Ka-i be the lowest group which still contains H. K K then does not contain all the substitutions of H. We apply now the formula B) of § 48 to the groups &T A + , , K K , and H, taking these in the place of G, K, , and H. We find P=/i+/i + ...+/-. This equation has two solutions, since the/'s, being divisors of h = p a , are powers of p: either we must have f\ =/ 8 = ... =1 and w —p, or else m = 1 and j\ = p. In the former case it would follow that h = d u i. e», that H is a subgroup of K K , which is contrary to hypothesis. Consequently/, = p, i. e. H and A" A have a common sub- group of order j) a ~\ This is the required group. * To Theorem X we can now add the following Corollary. Every group of order p a ■ p^ jp 2 aj . . . can be con- structed by the combination of subgroups, one of each of the orders p a ,Pi ai ,p 2 °*, . .. A smaller number of subgroups is of course generally sufficient. A further extension of the theory in this direction is not to be anticipated. Thus, for example, the alternating group of four ele- ments, which is composed of the twelve substitutions 1, (.C|.»'_,) (./" ..(', I. I .'Vl' ) I .''_,.«', ), (.!',. '') M •''_>•'': ). ' ' •*V*':i/5 ' ■' V j '•*%,)> V^1**V i '" '■'_■' \'i)) ' ' ' •' ' •' '. )i v^r^^y? v^W**3/j (.*'■','' ), has no subgroup of order 6. ^ 51. We insert here another investigation based on the con- struction of tables as in § 41. Let H be a group of order // affecting the n elements .r, , .»•_,, ...... ■ G. Frobenlus: Crelle 01. p.3£ MULTIPLE-VALUED FUNCTIONS —ALGEBRAIC RELATIONS. 55 From these n elements we arbitrarily select any k, as x u .'•_., . . . x , and let H' be the subgroup of H which contains all the substitu- tions of the latter that do not affect .*,, x 2 , . . . .<•, . Suppose In! to be the order of H', and /, = 1 L, . . . t,,' to be its several substitu- tions. We proceed then to tabulate the substitutions of H as fol- lows: Given any substitution s„ of H, suppose that this converts •<-,, .r,, . . . .<■,. into x aj ,x a , . . . x ah , in the order as written. Then all the substitutions also convert ,r, , .v., . . . x k into x\ x , x^, . . . -<'«,, respectively, and these are the only substitutions of H which have this effect. We take these various sets of h substitutions for the lines of our table, which is accordingly of the form 1, fc, t,, ■ • • *W i "2 j t 2 S-i , t 3 S%, , . fv s 2s T) s 3 , toS 3 , '3*3 J • • • 'It' 8-3) Sju, * - ,,S V > *3 S M J • . • *ft'V The substitutions of the table are obviously all different, and conse- quently p. h'»=h. Again, suppose that r, is any substitution of H which contains among its cycles one of order k, say (1) to 1 = (. a is contained in r„. If now we denote the substitutions of H which do not affect .c a . ,.<•„,, . . . a ■„, . i. e. those of the group 8 a ~ l H's a = H a by 1 / (a) / (o) f '(a) and by right hand multiplication by y„ form the line (3) v a , V a v a , then (3 ) contains all the substitutions of H which involve the cycle j(A^I, I tA. jtX,^t4-2/y \ - 4 3/5 (at* ry* ry* \ ( ry* ry* / Y* \ I ~Y* 'Y* 'Y* I I 'Y* Y* 'Y 1 \ the second set being non-conjugate with the first. § 52. We return now to the table constructed in § 45. This table did not possess the last of the four properties noted in § 41 ; the substitutions of one line were not necessarily all different from those of the other lines. For every group certainly contains the identical substitution 1, which therefore occurs /' times; and again in the example of § 46 three other substitutions occur in each of the three groups. We have now to determine in general when it is possible that one and the same substitution shall occur in all the groups G^G-,, . . . G p belonging respectively to the several values ft, Section Cf. Frobenius, Crelle CI. p. 273. followed by an article by the Author, ibid. CIII p. 321. tL. Kronecker: Monatsberichte d. Berl. Akad. IsT'J. 208. *iS THEORY OF SUBSTITUTIONS. If we apply to the series of functions y,,

~ l G, v, ...a~ 2 Gp; consequently we have 7 = H; that is, the group H is unaltered by transformation with respect to any substitution; it includes therefore all the substitutions which are similar to any one contained in it. We proceed now to examine the nature of a group H of this character. We consider in particular those substitutions of H which affect the least number of elements, the identical substitution excepted. It is clear that these can contain only cycles of the same number of elements, since otherwise some of their powers would contain fewer elements, without being identically 1. We prove with regard to these substitutions, first that no one of their cycles can contain more than three elements. For if H con- tains, for example, the substitution and if we take a— (x t x t ), then, since *~ *H 4, the substitution of H which affects the least number of elements cannot contain more than one cycle. For otherwise H would contain substitutions of the form and therefore the corresponding conjugate substitutions with respect to n — (.'•,.(•-) S a = (^V'j )'■''.■'':,)••• , $p = (•<',. <\.<'. | |.r ,■',.''. ) . . . Consequently the corresponding products 8 a ~ l 8 a '= O,) (X 2 ) (,C, ...)..., 8p~V = fa) ('<'-') fo) (• t 'r'V,). • ., which are not 1, but affect fewer elements than s, must also occur in H, which would again be contrary to hypothesis. If then n > 4, either H consists of the identical substitution 1, or if contains a substitution (■r tl ,r v ), or a substitution («r A ^V a \ )• In the second case H must contain all the transpositions, that is H is the symmetric group. In the third case H must contain all the cir- cular substitutions of the third order, that is H is the alternating group. (Cf. §§ 34-35). Returning from the group H to the group G, it appears that if G u G 2 , . . . G p have any substitution, except 1, common to all, then either the second or the third case occurs. H, which is contained in 6r, includes in either case the alternating group; G is therefore either the alternating or the symmetric group, and p = 2, or p = l. If, however, n = 4 we might have, beside s, = 1, another substi- tution in the group. With this its conjugates, of which there are only two, must also occur. The group H cannot contain any further substitu- tion without becoming either the alternating or the symmetric group. We have then the exceptional group 6<> THEORY OF SUBSTITUTIONS. H=[s 1 = 1, S 2 , s , 8 + ], and this actually does transform into itself with respect to every sub- stitution. Returning to the group G it follows from § 43, Theorem II. that the order of G is a multiple of that of H, that is, a multiple of 4; again from Theorem II the order of G must be a divisor of 4! = 24. The choice is therefore restricted to the numbers 4, 8, 12, and 24. The last two numbers lead to the general case already dis- cussed where ,"=2. or 1. The hrst gives G= H, /> = 6, and for example, c'j = (.(' r r_, T •••';;• I',) (•<'). *':; T ■ l ':-'\)i f: — \X{X 2 -J- #3X4) (-'Vi 1 •'.•■^3) C' I ./ j I 1 -~\— yCcpJCi f ( .I'j./'i - J X ...{ ;j ), C ^ \ .1 j,( ' .. — |— .1 .,.1 if \ ,t j.f ., I •';;*' j ) c- = ( ,c v i' i -\- .<'_.■'';; ) (.r|.r._, -\- .<•..<')),

T8 '"1 -X 4 I - ■ Theorem XI. If n > 4 Mere is ?io function, except the al- ternating and symmetric functions, of which all the p values are unchanged by the same substitution (excluding the case of the identi- cal substitution). Ifn = 4, all the values of any function belong ing to the same group with 2 • • • Pp) = v'ir'jV':; • • • V'p = ^p(Cn Co, . . .C ). the it's are the coefficients of an algebraic equation of which ft . ft, . . . values ft, ft, . . . ft, of a p-valued integral rational function

the coefficients of which are rational integral functions of the ele- mentary symmetric functions c n c,, ... c n of the elements x n x 2 , ... x'„. ^ hi. As an example we determine the equation of which the three roots are p, = x x X , + X a X t , ft - .r r r, + x. 2 Xi , ft = avr, -f x 2 x 3 , where a?,, .<•_,, cc 3 , sc 4 are themselves the roots of the equation /(.-•) = x 4 — e t x 3 + c 2 x 2 - c z x + c 4 = 0. We find at once ?i+?2 + ?3= SUv\) = r,; and again, by § 10, Chapter I, ', — ff 2 ~ 1 G 1 ff 2 l>elonging to = l is divisible by the product of /< %J — g factors x a - — Xp. But since J is symmetric in the ,r A 's, the presence of a factor ,r a — xp requires that of every other factor x y — x&, and consequently of J = // (.'- a — Xp)', the discriminant of f(x). Suppose that J' is a> the highest power of J which is contained as a factor in Jo, then, as J contains n{n — 1) factors x a — Xp, and consequently J' contains n(n — Y)t such factors, we must have n (n — !)*>/>[ g gj t > t M = 2 n(n — 1)' The number f can be only when g = ^ , that is, when all the transpositions occur in 6r,.

If (pis not symmetric, the exponent of J is not zero. If the group of and (x/j) 1 ^ are both symmetric, v'' p is symmetric also. We write then If (t'i be any root of this equation, and if to be a primitive (2p) lb root of unity, then all the roots are = n ! . All the values of a function are of the same type, and consequently there are substitu- tions which transform one into another. Suppose, in the case p — n\, that 2. The case n = -4 furnishes no exception. In this case the group common to all the values of o might be the special group (Theorem XIII) U" — |_-t) Kp^v^i) Kph'El)} (•'']•'':) ( • t '2 , l'i)i \p^V^i) ' '''j , ':i)J 5 c would then be a six-valued function, and there must be a substi- tution v which converts c x into wc, and which is of order 6. But there is no such substitution in the case of four elements. § 58. Finally we give a proof which is based on the most ele- mentary considerations and which moreover leads to an important extension of the theorem under discussion. In the first place we may limit ourselves to the case where ," is a prime number. For, if p=p-q, where p is a prime number, it follows from that there is also a function c'of which a prime power, the p xh , is symmetric. If, accordingly, we denote by

= 1 , and consequently, since p is a prime number, p = 2 and is multiple-valued, while its q tix power is two-val- ued, q being prime. Then there is some circular substitution of the third order it= (x a XpX y ), which does not occur in the group of , since, if this group contains all the substitutions of this form, it must be the alternating group (§ 35). Suppose, then, that 4'o j #1, but that 9 '. ff « = <.''!* = Si + $V^, since

4'ix Multiplying these three equations together and removing the func- tional values, we have MULTIPLE-VALUED FUNCTIONS AiGEBBAIC RELATION-. 67 If now we assume // > 4, then the group of 4' cannot contain all the circular substitutions of the fifth order, (Theorem X, Chapter II). If r is one of those not occurring in the group of 4, there is no multiple-valued function a power of which is two-valued, if the elements .>■ are independent quantities. § 59. We conclude these investigations by examining for u "^ 4 the possibility of the existence of functions having the property dis- cussed above. The case n = 2 requires no consideration. In the case n = 3 we undertake a systematic determination of the possible functions of the required kind. We begin with the type cr, = ax{ + ,ix., r + yx{, and attempt to determine «, ,5, y so as to satisfy the required condi- tions. For this purpose we make use of the circumstance that some a = (xjXtfc-^) converts c x into wc, (w 3 = 1) so that tp 9 — ax/ + (IxJ + yxf = «(«.r,' -j- {ix: + yx/), y = iua . t j = w^ = to a, a = w,j = wy = to a = a. The last three equations can be consistently satisfied for every value of a. We may take a — 1 ; and therefore is a function of the required type. This result is confirmed by actual calculation. We find 08 THEORY OF SUBSTITUTIONS. P 8 = 3 -..-. i + 6. <•,•.<•;.)•;-- — (.<■,-.<•/ + .'V'- a =toy>i, we have the series of equations CDjJ = U) ! 3 y — uj a = a , y x = oia, , /9, = ury 1 — ura 1 } a, = u){i x = ory^ = aj 8 a, = a, . All of these equations are satisfied independently of the values of a and « n and we have But again, the substitution r = (x^x^) converts cr, into cr T , where c T is equal to the product of ft by some cube root of unity, since MULTIPLE-VALUED FUNCTIONS ALGEBRAIC BELATION8. 01) V, 1 = (f T \ Whether this cube root is 1, a, or w 2 cannot be deter- mined beforehand. We find , we have then The function c { is therefore a combination of the three values of a function which we have already discussed. The group of -» — 11 -■>-» y _J fy* /vi — /i/ / y* 'y* I— / y* / y* — ■?/ For then cr, coincides with the expression obtained above for the case n = 3 ; and since y x , y, , y s , are the roots of the equation // — <'..'//" + (CiC-s—lCt) y—(crc i — ±c,e i + c;) = 0, where the c's are the coefficients of the equation of which the roots x x , x. 2 , x 3 , x i , (§ 54), we can translate the expression obtained for n = 3 directly into a two-valued function of the four elements x t , Xo, x 3 , x t , since we have (§ 54) J„ — J . CHAPTER IV TRANSITIVITY AND PRIMITIVITY. SIMPLE AND COMPOUND GROUPS. ISOMORPHISM. £ 60. The two familiar functions ' 1 ' ' I iA/Q.iJUa a tX.it/. i tA ■>*/, I differ from each other in the important particular that the group belonging to the former (rv* /vt /v» /y» \ / (i .» v» v. V I .( ].« ;:.( ._,.< j /, ( .1 [.( ,.( _..( ;. ) | contains substitutions which replace .r, by .<_,, oj 3 , or r, , while in the group belonging to the latter G-j = |_1, (■'']•'.')> ('*':!-* - 4)5 (•i*i-*' L ,) (■-l':i • • • x a+b> an( i s0 on, but none which, for instance, replace .»', by x a+ \ (A ^ 1), and so on. The maximum possible number of substitutions within the several sys- tems is a!, 6!, ... , and consequently the maximum number in the given group, if a, (>,... are known, is a! bl . . . If only the sum a -\- b -\- ...= >t is known, the maximum number of substitutions in an intransitive group of degree n is determined by the following equations : (n— 1)1 l! = ^=i(n — 2)! 2! >(n — 2)! 2!, (n > 3) (n— 2)! 2!=-^=^(w— 3)!3! > (n— 3)! 3!, («>5). o Theorem I. The maximum orders of intransitive groups of degree n are (w— 1)!, i(n— 1)!, (to— 2)!2!,(w— 2)!, (n — 3)!3!, (w— 3)!2!, . . . The first two orders here given correspond to the symmetric and the alternating groups of (n — 1) elements, so that in these cases one element is unaffected. The third corresponds to the combination of the symmetric group of (n — 2) elements with that of the re- maining two elements. The fourth belongs either to the combina- tion of the alternating group of (n — 2) with the symmetric group of the remaining two, or to the symmetric group of (n — 2) ele- ments alone, the other two elements remaining unchanged; and so on. The construction of intransitive from transitive groups will be treated later, ( § 99). § 61. We proceed now to arrange the substitutions of a transi- tive group in a table. The first line of the table is to contain all those substitutions S 1 := 1 , S> , S3 , . . . S m which leave the element x x unchanged, each substitution occurring only once. From the definition of transitivity, there is in the given group a substitution e 2 which replaces .r, by .<■_,. For the second line of the table we take "21 S 2 ff 2j N )' 7 J> • • • 8 m ff 2 . ,'1 THEORY OF SUBSTITUTIONS. We show then, 1) that all the substitutions of this line replace .r, by for every .s A leaves sc, unchanged and respectively; this could only be the transposition (x i x 5 ), and this cannot occur in the alternating group. In general, we can show that the alternating group of n ele- ments is always (n — 2)- fold transitive. The requirement that any (n — 2) elements shall be replaced by (n — 2) others may take any one of three forms. In the first place it may be required that (n — 2) given elements shall be replaced by the same elements in a different order, so that two elements are not involved. Secondly, the requirement may involve (n — 1) elements, or, thirdly, all the n elements. In the first case suppose that a is a substitution which satisfies the conditions, and let r be ih.e transposition of the two remaining elements. Then vr also satisfies the conditions, and one of the two substitutions t, i~ belongs to the alternating group. If (n — 1) elements are involved, suppose that the remaining element is x„ , so that neither the element which replaces x n nor that which x„ replaces is assigned. The elements which are to replace Xi , a? 2 ,...&„_] are all known with the exception of one. Suppose that it is not known which element replaces x n _ x . Then from the elements .r, , ,<•._,, . . , <£„_, we can construct one substitution which satisfies the requirements, say a — ( . . . x a x„ _ x x b ....)... , and from the n elements a second one, only distinguished from the first in the fact that ;r„_, is followed by x„, thus - = (....*•„.<•„,.<•.(„...) = t . (x b x„). Then either <> or - belongs to the alternating group. Finally, if all the n elements are involved, there are two elements for which the substituted elements are not assigned. Suppose these 1 4 THEORY ui' SUBSTITUTIONS. to be .*•„_, and .<■„. If now the elements are arranged in cycles in the usual manner, there will be two cycles which are not closed, the one ending with x n _ , , the other with x„ . We can then construct two substitutions a and r which satisfy the requirements, the one being obtained by simply closing the two incomplete cycles, the other by uniting the latter in a single parenthesis. From Chapter II, Theorem XI, it then follows that either t or - belongs to the alterna- ting group. The alternating group of n elements is therefore at least (n — 2)- fold transitive. It cannot be (n — l)-fold transitive, since it contains no substitution which leaves x 1 ,x 2 , . . . r„__, unchanged, and con- verts A'„_ l into .r„. § 63. If G is a /.-fold transitive group, the subgroup G' of G which does not affect .>', will be (k — l)-fold transitive; the subgroup G" of G' which does not affect x 2 will be (k — 2)-fold transitive, and so on. Finally the subgroup G( k ~ l > which does not affect x\, a? 2 , . . . £Ci_] will be simply transitive. Applying Theorem II successively to G' A_1) , . . . G", G', G, we obtain Theorem III. The order r of a k-folcl transitive group is equal to n(n —1) (n — 2) . . . (n — k-\-l)m, where m is the order of a a n subgroup which leaves k elements unchanged. $ 64. A simply transitive group is called non-primitive when its elements can be divided into systems, each including the same number, such that every substitution of the group replaces all the elements of any system either by the elements of the same system or by those of another system. The substitutions of the group can therefore be effected by first interchanging the several systems as units, and then interchanging the elements within each separate sys- tem. A simply transitive group which does not possess this property is called primitive. For example, the groups UT] — [_1, (•l' l .l' : ) J (./',.<•,), \X v X.i) (,'''/''|)i '•']•'':) (•'V*'4/> Kp^V^i) Kp^i^i/i \X\Xyl' s' \>i ' ■'']■' :'_■';' |5 ' — ) A, \X i X.2%i)i (•'']■'' '• I •' •' •' I? \XiXyT,,-!' _..(', '' T '' .''yC^j f GENERAL CLASSIFICATION OF GROUPS. 75 are both non-primitive. G x has two systems of elements, x x , <«■_, and The powers of a circular substitution of prime order form a primitive group, e. g. Gr 3 = [l, (.r, .*•_,.<•,), (.r,.r..r,( |. The powers of a circular substitution of composite order form a non-primitive group. If the degree of the substitution is 11 =Pi ai ■ i>> a2 ■ lh ai ■ • • , where p u p.,, p 3 , . . . are the different prime factors of u, the corresponding systems of elements can be selected in [('/, + 1) ( a 2 + 1) {' L .i + 1) • • • — 2 J different ways, as is readily seen. For example, in the case of the group UTj — |^1, (•'V'_"'':;'''e''.y''i./» ( ■'V';''".) ( ''VY'f'j (•'V'»' (•'_:•'':,) ' ■';•'■ )< we may take either two systems of three elements each, x l , x 2 , x 5 and ■"'■2i -''4» •*'.,? or three systems of two elements each, .r,,a' 4 , x 2 , x Tj , and .»■,, x 6 . A theorem applies here, the proof which may be omitted on account of its obvious character: Theorem IV. If, for a non-primitive group, the division of the elements into systems is possible in two different ways such that one division is not merely a subdivision of the other, then a third mode of division can also be obtained by combining into a new system the elements common to a system of the first division and one of the second. It must be observed that a single element is not to be regarded as a "system" in the present sense. Thus the group G t above admits of only two kinds of systems. >} 65. The elements of a non-primitive group G can be ar- ranged in a table, as follows. The first line contains all and only those substitutions •S] = I, s.,, s 3 , . . . s m which leave the several systems unchanged as units, and which accordingly only interchange the elements within the systems. (The line will of course vary with the particular distribution of the ele- ments in systems.) From the definition of transitivity, (for the names "primitive" and "non-primitive" apply only to simply transitive groups), there 76 THEORY 01 SUBSTITUTIONS. must be in the given group a substitution S 2 ff 2) S 8 ff 2j • • • S ,.^l- We show then, 1 ) that all the substitutions of this line produce the same rearrangement of the order of the systems as . The groups H', H", . . . are similar, for if t = (x' l x l ia} . . .). . . is a substitution of G, then the transformation t~ 1 G l t= G { will convert H' into H a \ The order of H' is a multiple of -- and a divisor of — !, where ," is the number of systems of non-primitivity. § 68. The following easily demonstrated theorems in regard to to primitive and non- primitive groups may be added here: Theorem VI. If from the element* .»-,, ,»•_,, . . . x„ of a tran- sitive group G any system x\, x'.,, . . . can be selected such that every substitution of G which replaces anyx' a by an x 1 p permutes tin x n s only among themselves, then G is a non-primitive group. Theorem VII. If from the elements x t ,x 2 c„ of a transitive, group G tivo systems .<■',, .*•'_,, . . . and .<•",, x".,, . . . can be selected such that any substitution which replaces any element ■<■'„ by an x H p replaces all the x f, s by .*•'"*, then G is a non primitive (/ roup. Theorem VIII. Every primitive group Q contains substi- tutions which replace an element x' a of any given system .*■',, x'.,, . . . by - The alternating group of n elements is commutative with every substitution of the same elements. 3) The group if of 2), being commutative with the symmetric group of the four elements x n x 2 , x :i , x t , is a self-conjugate sub- group of the latter. The alternating group of n elements is a self-conjugate sub- group of the corresponding symmetric group. Every group G of order r, which is not contained in the alterna- ting group A, contains as a self- conjugate subgroup the group H of order £r composed of those substitutions of G which are contained in A (Theorem VIII, Chapter II). The identical substitution is, by itself, a self-conjugate sub- group of every group. J> 7< ). We may employ the principle of commutativity to further the solution of the problem of the construction of groups begun in Chapter II (§§ 33-40). All substitutions of n elements which are commutative with any given substitution of the same elements, form a group. For if fj , t 2 . . . are commutative with s, it follows from that so that the product t x t 2 also occurs among the substitutions t. All substitutions of n elements which are commutative with a GENERAL CLASSIFICATION OF GROUPS. 81 given group G of the same elements form a group which contains G as a self -conjugate subgroup. For from £i — G t j = G, 1 2 — Gt-2 = G, follows (tit2)~ l G(t l Q = G; and among the V s are included all the substitutions of G. If two commutative groups G and H have no substitution, er^ept the identical substitution, in common, then the order of the smallest group K- \G,H\ is equal to the product of the orders of G and H. § 71. If a group G of order 2r contains a subgroup H of of order r, then H is a self-conjugate subgroup of G. For if the substitutions of H are denoted by 1, s.,, s 3 , . . . s r , and if t is any substitution of G which is not contained in H, then t, ts. 2 , ts 3 , . . . ts,. are the remaining r substitutions of G. But in the same way, t, s.,t, s 3 t, . . . s,.t are also these remaining substitutions. Consequently every substitution s a t is equal to some tsp, that is, wr have in every case t~ 1 8pt= s a , and therefore G~ l HG = H. If a group G contains a self-conjugate subgroup H and any other subgroup K, then the greatest subgroup L common to H and K is a self-conjugate subgroup of K. If the orders of G, H, K, L ci 1c are respectively g, h, k, I, then — is a multiple of — . iv V For if s is any substitution of K, then s~ x Ls is contained in K, since all the separate factors s ~ l , L, s are contained in K. But s~ l Ls is also contained in H, for L is a subgroup of H and s~ l Hs = H. Consequently s~ 1 Ls is contained in L, and, as these two groups have the same number of substitutions, we must have s~ x Ls = L, and L is a self- conjugate subgroup of K. The relation between the orders of the four groups follows at once from the formula of Frobenius (§ 48). We have only to take for the K of this formula the present group H, and to put all the di , d,, . . . d„, equal to I. We have then fr = ~r • hk I 6 82 THEORY OF SUBSTITUTIONS. .1 self-conjugate subgroup of a transitive group either affects every element of the latter, or els< it consists of the identical substi- tution alone. For if H = G~ l HG is a self- conjugate subgroup of the transi- tive group G, and if H does not affect the element .r, , then, since G contains a substitution S\ which replaces .<■, by .r A , it would follow that .s A _1 iJs A = ff would also not affect X\, that is, that //would not affect any element. If a self -con jugate subgroup of a transitive group G is intransi- tive, then G is non -primitive and H only interchanges the elements within the several systems of non-primitivity. For suppose that x l and .r A belong to two different systems of intransitivity with respect to H. Then G contains a substitution s K which replaces x\ by ;r A , and since s A ] Hs>, = H, it follows that S\~ } Hs K must replace .r x only by elements transitively connected with X\ with respect to H. But .s A ' replaces a* A by a?, and H re- places ,Ti by every element of the same system of intransitivity with '-, . Consequently the remaining factor s A must replace every ele- ment of the system containing .r, by an element of the system con- taining .r A . The systems of intransitivity of H are therefore the systems of non-primitivity of G. § 72. Another important property is that of the correspond- ence of two groups, of which an instance has already been met with in § 06. The two groups G and & of this Section were so related that to every substitution s of G corresponded one substitu- tion § of &, and to every 3 corresponded a certain number of s's. The correspondence was moreover such that to the product of any two .s's corresponded the product of two corresponding §'s. We may consider at once the more general type of correspond- ence, * where to every substitution of either group correspond a certain number of substitutions of the other, and to every pro- duct 8 a 8p corresponds every product « a ^ of corresponding S's and vice versa. We may then readily show that to every substitution of the one group correspond the same number of substitutions of the other. For if to 1 of the group G correspond 1, § 2 , § 8 , . . . §, of ©, * A.Capelli: Battagliul Glor. 1878, p. 32 aeq. GENERAL CLASSIFICATION 01 GROUPS. 83 then, if 3 corresponds to 8, all the substitutions §, §§ 2 , ) which is a self -conjugate sub- group of G (©). The correspondence of two groups as just defined is called iso- morphism. If to every substitution of G correspond q substitutions of ©, and to every substitution of 0) p substitutions of G, then G and © are said to be (p-q)-fold isomorphic, or if p and q are not specified, manifold isomorphic. If p = q = 1, the groups are said to be simply isomorphic. * Examples. I. The groups G = [1, (.r.-r,) (x 3 x b ) (x t X 5 ), (x v v,) (.r„r,) (.*',.»•,,), (.c^,) (x 2 X b ) {x 3 X 5 ), r=[l 5 (c^ 2 ), (~ 3 ), (? 2 | 3 ), (^ 8 j, (*,*,*,)] are simply isomorphic, the substitutions corresponding in the order as written. For if any two substitutions of G, and the corres- ponding substitutions of l\ are multiplied together, the resulting products again occupy the same positions in their respective groups. II. The groups G = [1, (.r v r,)l /'=[!, (4^ 2 ) (£&), (*,*) (5^,;, (f.fj (** 8 )] are (1 -2)-fold isomorphic. Corresponding to 1 of G we may take, beside 1, any other arbitrary substitution of /'. It follows that /' is simply isomorphic with itself in different ways. * (^1^3) of A an( i conversely to 1 of /' correspond 1, (iCiiCa) (x^) Of G. § 73. If G and F are (m-n)-fold isomorphic, then their orders are in the ratio of m : n. If L is a self-conjugate subgroup of G, and if A is the corres- ponding subgroup of I, then A is a self-conjugate subgroup of I. For from G~ 1 LG = L follows at once r~ l A r= A. In the case of (p-l)-fold isomorphism, it may however happen that the group A consists of the identical substitution alone. § 74. Having now discussed the more elementary properties of groups in reference to transitivity, primitivity, commutativity, and isomorphism, we turn next to certain more elaborate investigations devoted to the same subjects. The m substitutions of a transitive group G which do not affect the element x x form a subgroup G x of G. Similarly the substitutions of G which do not affect x., from a second subgroup G 2 , and so on to the subgroup G„ which does not affect x„. All those subgroups are similar; for if _2]+ . . .+[«]+ . . . +[2] + [0], where the symbol [1] does not of course occur, and [0] = 1. G l ,G. 2 ,...G„ therefore possess together n\n — 1] substitutions which affect exactly {n — 1) elements. These are all different, for any substitution which leaves only .r a unchanged occurs in G a , but cannot also occur in Gp. But this is not the case with substitutions GENERAL CLASSIFICATION OF GROUPS. 85 which affect exactly (n — 2) elements ; for if any one of these leaves both x a and x$ unchanged, it will occur in both G a and G p . Accord- ingly every one of these n\ii — 2] substitutions is counted twice, and G therefore contains £ n\n — 2] substitutions which affect exactly (n — 2) elements. Similarly every one of the n\if\ substitu- tions of q elements which occur in G x , G 2 , . . . G n is counted (n — q) times, and there are therefore only [q] different substitutions in G which affect exactly q elements. We have then for the total number of substitutions in G, which affect less than n elements » [ „_l] + | [ „_2]+... + -i- M +...+^[0]. If this number is subtracted from that of all the substitutions in G, the remainder gives the number of substitutions in G which affect exactly n elements. But from Theorem II r = ran = n[n — 1] -f- n[ii — 2] -(-...+ n\_q~\ + . . . + w [0]> and consequently the required difference N is .(|[»-2] + ?-[»-8] + - . . +.-==E=lw + • ■ ■ +^[0]> No term in the parenthesis is negative. The last one is equal to n—1 since [0] = 1. Consequently N > (n— 1). n Tlieorem IX. Every transitive gromp contains at least (n — 1) substitutions which affect all the n elements. If there are more than (n — 1) of these, then the group also contains substitu- tions which affect less than (u — 1) elements.* Corollary. A k-fold transitive group contains substitutions which affect exactly n elements, and others which affect exactly (n — 1), (n — 2), . . . (n — fc+1) elements. Those substitutions which affect exactly k elements we shall call substitutions of the k th class. We have just demonstrated the existence of substitutions of the n th , or highest class. If we consider a non- primitive group G, there is (§ 66) a second group © isomorphic with G, the substitutions of which interchange *C. Jordan: Liouville Jour. (2), XVII, p 351. SI) THEORY OF SUBSTITUTIONS. the elements .A,, A.,, . . . A„ exactly as the corresponding substitu- tions of G interchange the several systems of non-primitivity. Since 67 is transitive, 03 is also transitive. From Theorem IX fol- lows therefore Theorem X. Every nou-pri)>iitive group G contains substi- tutions which interchange all the systems of non-primitivity. § 75. We construct within the transitive group G the subgroup H of lowest order, which contains all the substitutions of the high- est class in G, and prove that this group H is also transitive. H is evidently a self-conjugate subgroup of G. If H were intransitive, G must then be non- primitive (Theorem VI). If this is the case, let 0) be the group of § 06 which affects the systems A } , A 2 , . . . A^ regarded as elements. 03 is transitive. To substitu- tions of the highest class in 03 correspond substitutions of the high- est class in G. (The converse is not necessarily true). Suppose that £) is the subgroup of the lowest order which contains all the substi- tutions of the highest class in ©. To .£) then corresponds eithor H or a subgroup of H. If ,£) is transitive in the A's, H is transitive in the x'b. The question therefore reduces to the consideration of the groups 03 and ^). £) can be intransitive only if © is non- primitive and G accordingly contains more comprehensive systems of non- primitivity. If this were the case, we should again start out in the same way from © and ,£), and continue until we arrive at a primitive group. The proof is then complete. Theorem XI. In every transitive group the substitutions of the highest class form by themselves a transitive system. § 70. Suppose a second transitive group G' to have all its sub- stitutions of the highest class in common with G of the preceding Section. If then we construct the subgroup H' for G', correspond- ing to the subgroup If of O f we have H' = H. Moreover the number IV, of the substitutions of the highest class in H is where ( r/), ;uul which is therefore of the form a — (a\) (.?,) . . . (.r 7 _ ,) («„«« . . .)• We have then (*- V)s _I = [(ajjJBj, ...)...(... a^as.)]* -1 = (& s _-i8«Pt)i and since this substitution affects only 3 elements, it follows that Secondly, suppose k_,) (x,, ...)...(.. .x q ). In the first case we take ffj = (a;,) (ajjj) . . . (^_i) («**« . . . ) (A; + 1) < x < g, and in the second *2 = 0»i) (#2) • • • (a»i-i) (a?»*A • . . •) ^ > g- It is evident that both are possible, if in the latter case it is remembered that n > q. We obtain then (r 2 _1 s a 2k — 2. Theorem XIV. If a k-fold transitive group contains any substitution, except the identical substitution, which affects less than (2k — 2) elements, it contains also substitutions which affectxxt y^ the most onlij three elements. This theorem gives a positive result only if k > 2. In this case, by anticipating the conclusions of the next Section, we can add the following (IKNKKAL CLASSIFICATION OF (iKOUPS. 89 Corollary. If a k-f old transitive group fc> 2 contains sub- stitutions, different from identity, which affect nut tuore limn (2k — 2) elements, it is either the alternating or the symmetric group. We may now combine this result with th i corollary of Theorem IX. If (I is fc-fold transitive, it contains substitutions of the class (it — k-\-l). Accordingly q^L(n — k-{-l). If G is neither the alternating nor the symmetric group, q > (2k — 2). Consequently M ( n — k + 1 ) > ( 21c — 2) and k 1 ) contains a circular substitution of three elements, it contains the alternating group. Suppose that s = (a;, x 2 x 3 ) occurs in the given group G. Then, since G is at least two-fold transitive, it must contain a substitution r, = (.r : ;) (.r v c r r A . ..)... and consequently also t — a l s(i = ^x 3 x i x p ), r~ 1 sr = (a; 1 M t ). In the same way it appears that G contains Consequently (§ 35) (/contains the alternating group. Theorem XVII. If a k-fold transitive group (k> 1) con fains a transposition, the group is symmetric. The proof is exactly analogous to the preceding. For simply transitive groups the last two theorems hold only 6a 90 THEORY OF SUBSTITUTIONS. under certain limitations, as appear from the following instances (x 2 — } -lj (•^'l^"2'^3/> v"4«'Vs,/j \*^7*^8*^9/> V^l'^5^'a^'^'«*^*7'^'3^4*^8/ ) * Both of these are transitive. But the former contains a substi- tution of two elements, without being symmetric, and the latter a substitution of three elements without being the alternating group. § 80. An explanation of this exception in the case of simply transitive groups is obtained from the following considerations. If we arbitrarily select two or more substitutions of n elements, it is to be regarded as extremely probable that the group of lowest order which contains these is the symmetric group, or at least the alternating group. In the case of two substitutions the probability in favor of the symmetric group may be taken as about f , and in favor of the alternating, but not symmetric, group as about \. In order that any given substitutions may generate a group which is only a part of the n ! possible substitutions, very special relations are necessary, and it is highly improbable that arbitrarily chosen substitutions s, = I J J ) should satisfy these conditions. The exception most likely to occur would be that all the given substitu- tions were severally equivalent to an even number of transposi- tions and would consequently generate the alternating group. In general, therefore, we must regard every transitive group which is neither symmetric nor alternating, and every intransitive group which is not made up of symmetric or alternating parts, as de- cidedly exceptional. And we shall expect to find in such cases special relations among the substitutions of the group, of such a nature as to limit the number of their distinct combinations. Such relations occur in the case of the two groups cited above. Both of them belong to the groups which we have designated as non-primitive. In (?, the elements x^ , .r_, form one system, and x 3 , x t another; it is therefore impossible that (V, should include, for example, the transposition (.r^). In G., there are three systems of non-primitivity x li X 2 i a hi ^r 4 , .^- , it*, ; , and «r 7 , .r„, .*•,,, (!, therefore cannot contain the substitution {x^x-,). GENERAL CLASSIFICATION OF GROUPS. 91 It is, then, evidently of importance to examine the influence of primitivity on the character of a transitive group, and we turn our attention now in this direction. § 81. With the last two theorems belongs naturally Theorem XVIII. If a primitive group contains either of the two substitutions it contains in the-fermer case the alternating, in the latter the sym- metric group. The proofs in the two cases are of the same character. We give only that for the latter case. From Theorem VIII, the given group must contain a substitu- tion which leaves ;*', unchanged and replaces x., by a new element x 3 , or which leaves x, unchanged and replaces x 1 by a new element .*•;, , or which replaces x x by x 2 or x, 2 by x t and the latter element in either case by a new element x 3 . If then we transform t with respect to this substitution, we obtain a transposition r' connecting either ac, or x, with x 3 , for example -' = (x^). The presence of - and t' in the group shows that the latter must contain the symmet- ric group of the three elements a?i,a? 2 ,£c 3 . From Theorem VIII there must also be in the given group a second substitution which replaces one of these three elements by either itself or a second one among them, and which also replaces one of them by a new element x t . Suppose this substitution to be, for instance, We obtain then -" = s~ l (x 2 x z ) s = (x^i), and it follows that the given group contains the symmetric group of the four elements x l , sc 2 , x 3 , x t \ and so on. § 82. We can generalize the last theorem as follows: Theorem XIX. If a primitive group G with the elements x y , .r_, . . . x„ contains a primitive subgroup H of degree k < n, then G contains a series of primitive subgroups similar to H, H\t Hn H21 • ■ ■ H„—i + 92 THEORY OF SUBSTITUTIONS. such that every 1I K affects the elements ■>\,.c zc*-h#*+a-ij where .r M .«\, . . . .<•/,_, may be selected arbitrarily. We take H x = H and transform H with respect to all the substi- tutions of G into i/, , H\ , H'\ , . . . Now let //', be that one of the transformed groups which connects the k elements x lt x 2 , . . . .r A of H } with other elements, but with the smallest number of these. We maintain that this smallest number is one. For if several new elements r, , =■,, ... occurred in H\, then from Theorem VIII there must be in the primitive group H\ a substitution which replaces one : by another ? and at the same timo replaces a second ? by one of the elements x 11 x 2 , . . . x k . Suppose that t— (f tt £/S • • • SyX S ...)••• is such a substitution, the case where /? = y being included. Then H'\ = tH l t~ 1 will still contain & y but will not contain = a . //",, there- fore contains fewer new elements : than H\. Consequently if H\ is properly chosen, it will contain only one new element, say .r, , , It will therefore not contain some one of the elements of H ] , say •'„. We select then from H x a substitution u = (. . . .r a x ti . ...)... and form the group u -1 H\u = H 2 . This group contains but not .*',,. In the same way we can form a group //., which affects only a?i, x 2 . . . x k _ 1 , x k+2 , and so on. It remains to be shown that a? n x 2 , . . . x k , can bo taken arbitra- rily, that is, that the assumption H = //, is always allowable. Sup- pose that //, contains x i , x 2 , . . . x k _ a . Then in the series //,, //_.,... there is a group //„ which also contains //, a + ] . Proceeding from //„ and the elements a;,, x 2 ,. . . X k __ a+1 , we construct a series of groups, as before, arriving finally at the group II. §88. Theorem XX. If a primitive group of degree h contains a primitive subgroup H of degree I: then G is at least (// — A; -J - 1 )-fold transit ire. From the preceding theorem //, affects the elements .r, , ,r. ; , . . . x t ; •//,,//.; the elements .»',,. i\, >; , .r, . , ; ;//,,//..//,( the ele- ments .<, . x . . . . ■•■ . .'■ . ,, r, ; ami so on. All these groups are GENERAL CLASSIFICATION OF GROUPS. 93 transitive; consequently, from Theorem XIII, \H U H 2 \ is two fold transitive, \H X ,H 2 , H t \ three-fold transitive, and finally /'= \H n . . . H„_,, + 1 \ is at least (n — k-\- l)-fold transitive. Therefore G', which includes /', is also at least (n — k-\- l)-fold transitive. * Corollary I. If a primitive group of degree n contains a circular substitution of the prime order p, the group is at least (n — p -\- l)-fold transit ire. For the powers of the circular substitution form a group H of degree p. Corollary II. If a transitive group of degree n contains a 2n circular substitution of prime order p < -=-, then, if the group o docs not contain the alternating group, it is non-primitive. From Theorem XV, every group which is more than [ -5- -f- 1 J- fold transitive is either alternating or symmetric. And since the presence of a circular substitution of a prime order p in a primitive group would require the latter to be at least (n — p-\- l)-fold tran- sitive, it would follow, if p < -~-, that the group would be more than (l+i> .fold transitive and must therefore be either alternating or symmetric. As these alternatives are excluded, the group must be non- primitive. § 84. In the proof of Theorem XIX the primitivity of the group H was only employed to demonstrate the presence of substi- tutions which contained two successions of elements of a certain kind. The presence of such substitutions would also evidently be assured if H were two-fold or many-fold transitive. Theorems XIX and XX would therefore still be valid in this case. The latter then takes the form : Theorem XXI. If a primitive group G of degree n con- * Another proof of this theorem is given by Rudio: Ueber primitive Gruppen, Crelle Oil, p. l. 94 THEORY OF SUBSTITUTIONS. tains a h-fohi transitive subgroup (&>.2) of degree q, then G is at l< ast (a — q-\-2) transitive. § 85. If the requirement that the subgroup H of the preceding Section shall bo primitive or multiply transitive is not fulfilled, the the theory becomes at once far more complicated. * We give here only a few of the simpler results. Tlicorom XXII. If a primitive group G of degree n con- tains a subgroiqj H of degree k < n, then G also contains a subgroup whose degree is exactly n — 1; or in other words: A transitive group G of n elements, which has no subgroup of exactly n — 1 elements, I 'nt has a subgroup of loiver degree, is non-primitive. Suppose that the subgroup H of degree ). < n affects the ele- n ments ;r, ,x.,,...X\. In the first place if X < - c y then the group G, on account of its primitivity, contains a substitution 8, which replaces one element of x x , ,r_, , . . . x K by another element of the same system and at the same time replaces a second element of x x , x,, . . . x K by some new element. Then H' = 8{~ 1 Hs x contains beside some of the old elements, also certain new ones, so that H , — \H,H'\ affects more than / elements, but less than n, since H and H' together n affect at the most (2 X — 1) < n. If the degree /., of H x is still < -jr-, n we repeat the same process, until /, is equal to or greater than ~. a Suppose that the elements of the last H x are x } , a\,, . . . .r A . Then the primitive group G must again contain a substitution s a which replaces two elements not belonging to //, by two elements, one of which does, while the other does not, belong to 7/, . Then the group H\ = s^H^s.r 1 will connect new elements with those of H x ; but, from the way in which 8 2 was taken, one new element is still not con- tained in H\ . That some of the old elements actually occur in H\ follows from the fact that /, > \ n. Accordingly H 2 = \H X , H\ \ con- tains more elements than H x but less than G. Proceeding in this way, we must finally arrive at a group K which contains exactly (n — 1) elements. •C.Jordan: Lionville Jour. (2)XVI. B. Marggraf : Ueber primitive Gruppen mlt transitlven Untergruppen geringeren Grades; Glessen Dissertation, i^iio. GENERAL CLASSIFICATION OF GROUPS. '.)"> If H is transitive, then H', and consequently H X =\H', H\, and so on to K, are also transitive. From Theorem XIII, G must therefore in this case be at least two-fold transitive. We have then the following Corollary. // a primitive group G contains a transitive subgroup of lower degree, then G is at least two-foul transitive. § 86. We turn now to a series of properties based on the the- ory of self- conjugate subgroups. Let H= [1, s 2 ,8 8 > • • •$»] be a self -conjugate subgroup of a group G of order n — km. The substitutions of G can be arranged (§ 41) in a table, the first line of which contains the substitutions of H. s, = l > S 2j S 3> . . . o m , a 2, 2 2 5 Ss7p, that is, the line of the table in which the product (s\T a ) (s^tp) occurs depends only on = / ' M , G^ : 1 = /', the order and the degree of every I' a is equal to e a (a = 1, 2, — « + 1 ). All the groups /'„ are simple. For 1\ is (l-r tt )-fold isomorphic with G a ^ x , and to the identical substitution in / '„ corresponds G a in G tt _,. Consequently, if J' a contains a self-conjugate subgroup dif- ferent from identity, then the corresponding self- con jugate sub- group of G a -i (§ 73) contains and is greater than G a . The latter would therefore not be a maximal self-conjugate subgroup of (?„_,. The groups / ', which define the transition from every G a to the following one in the series of composition, are called the factor groups of G. * *0. Holder; Math. Ann. XXXIV, ]>. 30 II. GENERAL CLASSIFICATION OF GROUPS. 97 § 88. Given a compound group G, it is quite possible that the corresponding series of composition is not fully determinate. It is conceivable that, if a series of composition G, Gi , Go , . . . Gp. , 1 has been found to exist, there may also be a second series Gr, G i , Gr 2 , . . . G i, , 1 in which every G' is contained as a maximal self-conjugate subgroup in the preceding one. We shall find however that, in whatever way the series of composition may be chosen, the number of groups G is constant, and moreover the factors of composition are always the same, apart from their order of succession. Suppose the substitutions of G x and G\ to be denoted by s a and s' a respectively. Let r x = r :e 1 be the order of Gr, , and r\ = r : e' x that of Gr',. The substitutions common to Cr, and G\ form a group J* (§44), the order x of which is a factor of both i\ and r\. We write r ] =xy, r\ = xy'. The substitutions of T we denote by »3> ■ • • ""..; r, •^' 7 l- ^i' 7 :-, §2 ff S» • • • §2 ff *; M"> §yff 1 , §yff 2 , § y ff 3 , . . . %y\«d)G= G- l SsG ■ G~ i 5\^G = s a s'p = * y t = M>«- The group ® is more extensive than Cr, or 6r'; it is contained in G; consequently, from the assumption as to G x and G' , & must be identical with G. The order of & is equal to xy ■ y'. For, if 5 a *Vv = M'j4*> it is easily seen that a = «, b ' = ft, c = y. Consequently the order of G is also xyy', and since we have r = ?',e, = xye t , r = r\e\ = xy'e\ it follows that 2/'=e,, 2/ = e',. This last result gives us for the order of l\ x r r, r ,1 1 < 1 I 1 1 1 We can show, further, that F is a maximal self- conjugate subgroup of G\ and of G\ , and consequently occurs in one of the series of composition of either of these groups. For in the first place 1\ as a part of G\ , is commutative with G,, and, as a part of (?,, is com- mutative with G\ , so that we have Gr i rG 1 = G\, G' 1 } l'G' ] = G l . But since the left member of the first equation belongs entirely to Gj , the same is true for the right member, and a similar result holds for the second equation. Consequently GENERAL CLASSIFICATION OF GROUPS. 99 G l - 1 rG l = r, G'r l i'G' l = r. Again there is no self-conjugate subgroup of G 1 intermediate be- tween Gi and /' which contains the latter. For if there were such a group H with substitutions t a , then it would follow from A) that t a ~ 's'p - l t a s'p — ffy, s'|3~ t a s'p = t a ■ t„r s'pT t a s p = t a v y = t& , that is, H is also commutative with G\. And since G x and G\ together generate G, it appears that H must be commutative with G. If now we add to the f a 's the §' 2 , §' 8 , . . . , then the substitu- tions %' a tp form a group. For since /' is contained in H and in G x , we have from A) This group is commutative with G, since this is true of its compo- nent groups H and (?',. It contains G\, which consists of the sub- stitutions §' a ffjs. It is contained in G, which consists of the substitu- tions §' a §j3«r y . But this is contrary to the assumption that G\ is a maximal self-conjugate subgroup of G. We have therefore the fol- lowing preliminary result: If in two series of composition of the group G, the groups next succeeding G are respectively G x and G\, then in both series ice may take for the group next succeeding G x or G\ one and the same max- imal self -conjugate subgroup 1\ which is composed of all the substi- tutions common to G x and G\. If e x and e\ are the factors of composition belonging to G x and G\ respectively, then V has for its factors of composition, in the first series e\, in the second e x . § 89. We can now easily obtain the final result. Let one series of composition for G be 1) G, Gi, G 2 , G 3 , . . . , r, ?-! = r: e, , r, = i\ : e 2 , r 3 = r,: e 3 , . . . , and let a second series be 2) G, G\, G' 2 , G' 3 , . . . , r, r' l = r:e\, r',-r\:e',, r\ = r', :e\, . . . Then from the result just obtained, we can construct two more series belonging to G: 100 THEORY OF SUBSTITUTION-. 3) G, ' i '■ <'i • • • 5 and apply the same proof for the constancy of the factors of com- position to the series 1) and 3), and again 2) and 4), as was employed above in the case of the series 1) and 2). The series 3) and 4) have obviously the same factors of composition. The problem is now reduced, for while the series 1 ) and 2) agree only in their first terms, the series 1) and 3), and again 2) and 4), agree to two terms each. The proof can then be carried another step by constructing from 1) and 2) as before two new series, both of which now begin with G, G { : 1') G,G 1} G 2 , &,$,%..., 3') Q,G lt r, ©,£>,&..., r, r,, r r 2 = r 1 :e' 2 , r" :i = r' 2 :e 2 , . . . These series have again the same factors of composition, and 1') and 1 ) and again 3') and 3) agree to three terms, and so on. We have then finally Theorem XXIII. If a compound group G admits of two different scries of composition^ the factors of composition in the two ruses are identical, apart from their order, and the number of groups in the two series is therefore the same § 90. From § 8S we deduce another result. Since G~ l TG belongs to G u because G~ l G t G = (?,, and also to G\ because (,' l G\G = G\, it appears that G~ 1 rG, as a common subgroup of Cr, and G\, must be identical with /', so that /'is a self-conjugate subgroup of G. From § 80 it follows that it is possible to con- struct a group Q of order e,e', which is (1 T )-fold isomorphic e x e | with G, in such a way that the same substitution of il corresponds to all the substitutions of G which only differ in a factor n. We will take now, to correspond to the substitutions 1, § 2 , §j, . . . §/ y of (?,, the substitutions ],<»,.(»., . . . w, -, of il, and, to correspond to the 1, §' z , *'■,, . . . »',., of 6r'j, the substitutions 1, "/,, «/,, . . . <-/, , of Q. In no case is %' a = %p% y , for the t's form the common subgroup of G x GENERAL CLASSIFICATION OF GROUPS. 1<>1 and G'i . Consequently the w's are different from the w' 's. Both classes of substitutions give rise to groups : fl, = [1, w 2 , a» 8 , . . . w,-J, P-o = [1, w' 2 , w' 8 , . . . w',.J, and, since § a §'/3 = ^V-v^ ^ follows that fy fl^ = Q\ Q x . Moreover every s in G is equal to %a&p 4, of the alternating group and the identical substitution. The corresponding factors of com- position are therefore 2 and hn\ The alternating group of more than four elements is simple. We have already seen that the alternating group is a maximal self-conjugate subgroup of the symmetric group. It only remains to be shown that, for n > 4, the alternating group is simple. The proof is perfectly analogous to that of § 52, and the theorem there obtained, when expressed in the nomenclature of the present Chap- ter, becomes: a group which is commutative with the symmetric group is, for n > 4, either the alternating group or the identical sub- stitution. It will be necessary therefore to give only a brief sketch of the proof. Suppose that H x is a maximal self -con jugate subgroup of the alternating group H, and consider the substitutions of H x which affect the smallest number of elements. All the cycles of any one of these substitutions must contain the same number of elements (§ 52). The substitutions cannot contain more than three elements in any cycle. For if H contains the substitution S — (J)CiX.^)C' i X i ...}..., and if we transform s with respect to ) (a^)]; 5) ^ = L The exceptional group G 2 is already familiar to us. § 93. We may add here the following theorems: Theorem XXYI. Every group G, which is not contained in the alternating group is compound. One of its factors of com- position is 2. The corresponding factor group is [(1 , z ] , z,, )]. The proof is based on § 35, Theorem VIII. The substitutions of G which belong to the alternating group form the first self-con- jugate subgroup of G. Theorem XXVII. If a group G is of order p a , p being a prime number, the factors of composition of G are all equal top. The group K of order p f obtained in § 30 is obviously, from the method of its construction, compound. It contains a self- conjugate subgroup L of order p f ~ l and this again contains a self- conjugate subgroup .1/ of order p f ~" ~, and so on. The series of composition of K consists therefore of the groups K, L, M, . . . Q, R, . . . o, 1, of orders P r \p f ~\P f ~\ ■ ■ ■P k ,P k '~\ • • -P, 1- The last corollary of § 49 shows that we need prove the present theo- rem only for the subgroups of K. If G occurs among these and is one of the series above, the proof is already complete. If G does not occur in this series, suppose that R is the first group of the series which does not contain G, while G is a subgroup of Q. We ■ apply then to G the second proposition of § 71. Suppose that H is the common subgroup of R and G. Then if is a self- conjugate subgroup of G, and its order is a multiple of p a ~ l and is conse- 10 1 THEORY OF SUBSTITUTIONS. quently either p° _I or p a . The latter case is impossible since then G would be contained in E. Consequently H is of order p a ~ ', and the theorem is proved. Theorem XXVIII. If a grows G of order r contains a v self -conjugate subgroup H of order — then no substitution of G, e which does not occur in H can be of an order prime to e. * We construct the factor group F = G : H of the order e. No one of the substitutions of /', except the identical substitution, is of an order prime to e. To any substitution s of G which does not occur in H corresponds a a which is different from 1. On account of the isomorphism of G and r, there corresponds to every power s* of s the same power T)(. For in the series preceding J we may assume the sequence if„_i ,a ', \H v _ l (a) , H v ^^ ] \, . . . tooccur. Accordingly wemust have GENERAL CLASSIFICATION OF GROUPS. 107 or {H v _^y\H v _^)-'H v _^H v _^H v _^ Corollary IV. The last actual group M of the principal series of G is composed of one or more groups similar to one another, which have no substitutions except identity in common, and which are commutative with one another. § 95. We have now to consider the important special case where e is a prime number p. Instead of H' v _ l ,H" v _ l , ... we employ now the more conven- ient notation H', H", H'", . . . flW. Then H' is obtained from J by adding to the latter a substitu- tion t Y , the p th power of which is the first to occur in J. We may write (§ 91) H' = tfJ, H"=tfJ, H'"=tfJ,... (« = 0, 1, ...p — 1). Since J" is a self-conjugate subgroup of every one of the groups H', H", . . . ,we have t 1 -«jt 1 a = j, u- a Jh a = J, t 3 - a Jt i a = J,... and, if we denote the substitutions of J by i x , i, ,%,... , . t — a-i — 1/ a — A i — a," — •/ a — A f — a/ — 1/ a — A ti a h = hV (hh) . W = hk a (hh) , U a h = iA a ihh), • • • . that is, the substitutions of H', of H", of H'", and so on, are com- mutative among themselves, apart from a factor belonging to J. Since we can return from J to If by combining the substitutions of H' and H", for example, into a single group (§ 88), we have from § 94, Corollary III t.r %t 2 = *,"*,, tr %~ % = t./i, , and consequently, by combination of these two results, t l -%-'t l t, = t.m, =t./ +i i 3 , t--% = t/ + %. The left member of the last equation is a substitution of H', the 1< IS THEORY OF SUBSTITUTIONS. right member a substitution of H". Since these two groups have only the substitutions of J in common, the powers of /, and /._, must disappear. Consequently « = 1, /? = — 1, and V (t l i ] )(tX) = (tjjit^i.,, (t 1 H 1 )(t 2 H 2 ) = (t a H 2 )(t 1 H 1 )i i . The substitutions of the group formed from J, t t , and t, are therefore commutative among themselves, apart from a factor be- longing to J. The same is true of the group formed from J, t t , and t,, or from J, t._,, and t :i , and consequently of the group \J, t x ,L, t 3 \, and so on, to the group H itself. (It is to be noted that Corollary III of § 94 involves much less than this. There it was a question of the commutativity of groups, here of the single substitutions. ) Every two substitutions of H are, then, commutative apart from a factor belonging to J. We will prove now the converse proposi- tion: If two substitutions of H are commutative apart from a fac- tor belonging to J, then £ is a prime number. In fact this will be the case, if the substitutions of H' have this property. For, this being assumed, if z were a composite number, suppose its prime fac- tors to be q, q, q", . . . We select from H\ , in accordance with Theorem XXIV, § 01, a substitution t which is not contained in J. The lowest power of t which occurs in J will then be, for example, P. Transforming, we have H'- l (t a J)H'=H'- l t a H'H'- l JI[' = H'-H a H'J, and, since by assumption, t a H' — H't a J, H'- l (t«J)H' = t«J. The group ]t, J{ is therefore a self-conjugate subgroup of H', which contains J and is larger than J. Moreover, it is contained in H', and is smaller than H'. For, if / is commutative with J, then from §§ 37-8 the order of \t, J\ is r"q < ?*"e. This is contrary to the assumption that J is the group immediately following H' in the series of G. GENERAL CLASSIFICATION OF GROUPS. 10& Theorem XXX. If, in the principal series of composi- tion of G, the order r of H is obtained from the order r" of J by mult i plication by p v , where the prime number p is the factor of com- position for the intervening groups in the series of G, then the substitutions of H are commutative among themselves apart from factors belonging to J. Conversely, if this is the case, the factors of composition of the groups between H and J are all equal to the same prime number p. § 96. We turn finally to certain properties of groups in rela- tion to isomorphism. If L is a maximal self-conjugate subgroup of G, and A the corresponding group of /', then A is also a maximal self -conjugate subgroup of r. For if /' contained a self-conjugate subgroup 0, which con- tained .1, then the corresponding group T of G would contain L. The series of composition of G corresponds to that of /'. If { i and F are simply isomorphic, all the factors of the one group are equal to the corresponding factors of the other. But if G is mul- tiply isomorphic to I\ then there occur in the series for G, besides the factors of V, also a factor belonging to the group S which cor- resjionds to the identical substitution of F. The proof is readily found. If G is multiply isomorphic to I', then G is compound, and S is a group of the series of composition of G. § 97. Suppose that G is any transitive group of order r, affect- ing the n elements .r n x,, . . . x lt . We construct any arbitrary n\- valued function I of x : , x 2 ,, . . . x„, denote its different values by r , , f> , . . . ':„ < , and apply to any one of these, as r : , all the substitu- tions of G. Let the values obtained from r, in this way be The r substitutions of G will not change this system of functions as a whole, but will merely interchange its individual members, produ- cing r rearrangements of these, which we may also regard as sub- stitutions. These substitutions of the I's, as we have seen, form a new group /'. The group r is transitive, for G contains substi- 110 THEORY OF SUBSTITUTIONS. tutions which convert I, into any one of the values ?j , £ a , . . . f r , and therefore the substitutions of /' replace £, by any element ?,,€ s , ...£ r . Again every substitution of Gr alters the order of ■?!,!_., . . . ^,., for I is a n\- valued function. Consequently every sub- stitution of /'also rearranges the ?i,£ a ,...£ r . The order of T is therefore equal to its degree, and both are equal to r. G and /' are simply isomorphic. For to every substitution of G corresponds one substitution of /', and conversely to every substitu- tion of /'at least one substitution of G. And in the latter case it can be only one substitution of G, since G and /'are of the same order. Theorem XXXI. To any transitive group of order r cor- responds a simply isomorphic transitive grouj), the degree and order of which are both equal to r. Such groups are called regular. § 98. Theorem XXXII. Every substitution of a regular group, except the identical substitution, affects all the elements. A regular group contains only one substitution which replaces a given element by a prescribed element. Every one of its substitutions coyisists of cycles of the same order. If two regular groups of the same degree are {necessarily simply) isomorphic, they are similar i. e., they differ only in respect to the designation of the elements. Every regular group is non-prim it ire * The greater part of the the theorem is already proved in the preceding Section, and the remainder presents no difficulty. We need consider in particular only the last two statements. Suppose that /', with elements ? x , £ 2 , . . . ?„ and substitutions • • • s '« > the isomorphism being such that to every 4 , and so on to a m and c,„. If we apply all the substitutions of /' to the system of values rii r2> • ■ »r«) we obtain rearrangements which can be regarded as substitutions of the new elements ■>) and consequently t a ^ = — 1, we write tbe corres- ponding equations: ',+ 4'* + — 2 equations of the system S) successively by the undetermined quantities y , //,, >j, . . . y p - 2 , and the last equation by y p _, = 1, and add the resulting products, wri- ting for brevity l i y P -x<; p l -ry P -i9 p ~ i +y P -s

t /. (?«) + . . . 4' P x M = A>y* -f am + ^2 + • . . . . . + ^-p-:Up -2-\-A p _ l y p _ 1 . From this equation we can eliminate c\,, <.'\ ; , . . . 4' P and obtain c'',. For this purpose we need only select the f/'s so that we have simul- taneously x(n) = 0, z(f,) = 0, ... /(c P ) = (); z("ft)+& Ill) THEORY OF SUBSTITUTIONS. In Chapter III, £ 53, we have shown that p,, c :j . . .

-V(c) = 0, the coefficients of which are rational in '*,,<:■_,, . . . c„. Again, the quotient X(-t = - V ■ ••!/>, = ±.'p- Or, if we write A'(cr) = crP — a, ? P" ' -f a.,c>> *—'... ± a p , we have ^\-=9"- 1 +[.,X'(< Pl )=R{< P ,\ *i = w^' The value of l} and con- sequently in terms of c^c.,, . . . c„ and p lt If now we multiply numerator and denominator of the expression for 4>\ in (2) by the product (3), we obtain (4) * = ^>. The denominator of this last fraction is rational and integral in c, , c, , . . . c„ ; the numerator is rational and integral in c, , c, , . . . c„ and c, . If the numerator -R^c'i) is of a degree higher than p — 1 with respect to c,, a still further reduction is possible. For suppose that where Q(c) and R,( , cV. , . . . c'' p . We have therefore Theorem II. If two p -valued functions — 1, with coefficients ivhich are integral and rational in c n c,, . . . c H * § 105. The converse of Theorem I is proved at once: Theorem III. If two functions can be rationally cr. pressed one in terms of the other, they belong to the same group. *Cf. Krouecker: Crelleoi. p. 307. lis THEORY OF SUBSTITUTIONS. In fact, given the two equations it appears from the former that , while from the latter equation it appears in the same way that the group of c'' contains that of y. The two groups are therefore iden- tical. Remark. Apparently the proof of this theorem does not involve the requirement that c and shall be rational functions. It must however be distinctly understood that this requirement must always be fulfilled. For example, in the irrational functions the expressions under the square root sign are all unchanged by the transposition a = {x 1 x 2 ). But it remains entirely uncertain whether the algebraic signs of the irrationalities are affected by this substitution. Considerations from the theory of substitutions alone cannot determine this question, and accordingly the sphere of appli- cation of this theory is restricted to the case of rational functions. If, in the last two irrationalities above, the roots are actually extracted and written in rational form ±{x 1 — x 2 ), ±(x 1 + x 2 ), it appears at once that the transposition a changes the sign of the former expression but leaves that of the latter unchanged, while in the case of the first irrationality this matter is entirely undecided. $ 106. Theorems I and III furnish the basis for an algebraic classification of functions resting on the theory of groups. All rational integral functions which can be rationally expressed one in terms of another, that is, which belong to the same group, are regarded as forming a family of algebraic functions. The number ji of the values of the individual functions of a family is called the order of the family. The several families to which the different values of any one of the functions belong are called conjugate families.* * L. Kronecker: Monatsber. d. Her\. Akad.. L879, p. 212. FUNCTIONS BELONGING TO THE SAME f4ROUP. 119 The product of the order of a family l>y the order of the cor- responding (/roup is equal to u\, where n is the degree of the group. Every function of a family of order p is a root of an equation of degree />, the coefficients of which are rational in c n c 2 , . . . c n . The remaining <> — 1 roots of this equation are the conjugate func- tions. The groups which belong to conjugate families have, if p > 2, n > 4, no common substitution except the identical substitution. For p = 2 tlie two conjugate families are identical. For [> = Q, u = 4 there is a family which is identical with its five conjugate families. § 107. In the demonstration of § 104 the condition that c and should belong to the same family was not wholly necessary. It is only essential that (p shall remain unchanged for all those substitu- tions which leave the value of c unaltered. ' The demonstration would therefore still be valid if some of the values of c'* should coincide; but the values of is said to be included in the family of the function c. <.'• can be rationally expressed in terms of . From the preceding considerations we further deduce the fol- lowing theorems: 120 THEORY OF SUBSTITUTIONS. Theorem V. It is always possible to find a function in terms of which any number of given functions can be rationally expressed. This function can be constructed as a linear combina- tion of the given functions. Its family includes all the families of the given functions. Thus any given functions cr, 0, /, . . . can be rationally expressed in terms of m =a

being accordingly a n '.-valued function. In this case every function of the n elements x t , x 2 , . . '. 0C„ can be rationally expressed in terms of <», and every family is contained in that of w. The family of <" is then called the Galois family. Theorem VI. Every rational function of n independent elements x l ,x 2 , . . . x„ can be rationally expressed in terms of every nl-valued function of the same elements: in particular^ in terms of any linear function -valued function 0, and if 9\*9l1 ■ ■ ■ 9« are the in values ivhich

x ), where the A's are rational, but in general not integral functions of (p t . We obtain therefore the equation (A,)

+ A 2 fa,) ?— •-' + . i . ± A m fa) = 0, of which the roots are \ of t'. The denominators of the A A 's and, in fact, their least common denominator is always a divisor of the discriminant J c , as appears from the proof of Theorem II. If (j.' is a symmetric function, there is no longer a discriminant, and the denominator is removed, as we have seen in Chapter III, § 53. § 109. One special case deserves particular notice. If the included group H of the function — if,, The at different values cr,, cr_,, . . .

, is not merely contained in the group G of 4'\ but is a self-conjugate subgroup of G, the family of t' 1 , is called a self-conjugate subfamily of the family of cr, . Theorem VIII. In order that all the roots of the equation (Ax) should be rationally expressible in terms of any one among them, as cr, , it is necessary and sufficient that the family of <.'-, should be a self -conjugate subfamily of that of cr,, i. e., that the groti)) of v, should be a self -conjugate subgroup of that of t'-, . The groups of cr,, cr_,, . . . c„, are then coincident. We consider in particular the case where m is a prime number. Suppose G?] to be the group of 4'i and Hi that of cr, . Since every substitution of (?, produces a corresponding substitution of the val- ues Cj, 9u ■ ■ • 9mi the group Cr, is isomorphic with a group of the cr's. The latter group is transitive and of degree m. From The- orem II, Chapter IV, its order is divisible by m, and from Theorem X, Chapter III, it therefore contains a substitution of order m. FUNCTIONS BELONGING TO THE SAME GBOUP. 123 For /// elements, where m is prime, there is only one type of such substitutions t = (ft ft ■ • • ft.)- The corresponding substitution r of G, therefore permutes ft> ftj • • • 'r.n cyclically. Moreover, since -'" corresponds to /"', it fol- lows that -'" leaves all the functions

mp i> n ! essarily contained in Gr, , which, being itself of order — , cannot P contain any other substitutions. From this it appears again that r is commutative with H x . Theorem IX. If the equation (A,) is of prime degree m, and if the group H x of tp x is a self -conjugate subgroup of the group G-'i of ft, then G x contains a substitution r which permutes y>i, '" _ Vi evidently all belong to the same group. It is therefore necessary that tyj should be a self -conjugate subgroup of G x . We proceed now to show conversely that, if the group if, is a self-conjugate subgroup of (?, , then a function y A belonging to H } can be found, the m th power of which belongs to (?, . Denoting any primitive m th root of unity by w, we write Xl = ft + <»?> + "V:; ■+-...+ «" " V« • If we apply to this expression the successive powers of t or r, we obtain J 2 \ THEORY OF SUBSTITUTIONS. /.■ = 9a + to 9» + '"'fi + • • • -f <"'" " ~Vl = «» '/l, /: — 9* + ">9i + <"V:. + • ■ ■ + "> m - Va = "'~ a ^i) consequently We have now to prove 1) that y x belongs to the group H t , and 2) that ■/_{' belongs to the group Gr, . In the first place, since But we may assume the function ^, to have been constructed by the method of § 31 as a sum of - terms of the form xfxfi . . . with ° mp undetermined exponents. The systems of exponents in c,, c,,, . . . c,„ will then all be different, and therefore, since the x's are independ- ent variables, the equation 9i + 9i 3 =92 + 9ii can hold only if tp x — i =

■ }> ] ■ p, . . . p v - valued function belonging to G v from a p-valued function belonging to G by the solution of a series of binomial equations. The latter are then of degree p 1} p 2 ,P3, ■ ■ •/>>■- respectively. 8 111. In the expression of a given function in terms of another belonging to the same family, we have met with rational fractional forms the denominators of which were factors of the dis- criminant of the given function. If we regard the elements .«•,, .<•,. . . . x„ as independent quantities, as we have thus far done, the discriminant of auy function

k V — 1 (/■ = 1, 2, . . . n >. then we can take "■' =P + <1 >/--l in such a way that all the >/ ! quantities **a = and q are entirely arbitrary, that "'a =m Ki !I-k — !>k- In fact we can, for example, take p = q- and q so large that even special values of q satisfy the conditions. Suppose then that the cVs are arranged in order of the magni- tudes of their moduli mod. 4'\ +i) • "We take then the integer e so great that cV > ( has <> distinct values, and consequently -L is not zero. CHAPTER VI THE NUMBEK OF VALUES OF INTEGRAL FUNCTIONS. § 112. Thus far we have obtained only ocasional theorems in regard to the existence of classes of multiple-valued functions. We are familiar with the one- and two-valued functions on the one side and the »!-valued functions on the other. But the possible classes lying between these limits have not as yet been systematically exam- ined. An important negative result was obtained in Chapter III, £ 4:2, where it was shown that p cannot take any value which is not a divisor of n\. Otherwise no general theorems are as yet known to us. We can, however, easily obtain a great number of special results by the construction of intransitive and non- primitive groups. But these are all positive, while it is the negative results, those which assert the non-existence of classes of functions, that are pre- cisely of the greatest interest. The general theory of the construction of intransitive groups would require as we have seen in § 101, a systematic study of iso- morphism in its broadest sense. We shall content ourselves there- fore with noting some of the simplest constructions. Thus, if there are n = a -f- b -f- c + . . . elements present, and if we form the symmetric or the alternating group of a of them, the symmetric or alternating group of b others, and so on, then on multiplying all these groups together, we obtain an intransitive group of degree n and of order r = e a!6!c! . . ., where e = l, I- \ ■ |i ■ ■ ■ > according as the number of alternating groups employed in the construction is 0, 1, 2, 3, . . . , the rest being all symmetric. For the number of values of the corresponding functions we have then n\ '' ~ *a\b\c\...' a — 5; a — 4, 6 = 1; a = 4, 6=1; a = 3, 6 = 2; a = 3, 6 = 2; a = 3, 6 = 2; THE NUMBER OF VALUES OF INTEGRAL FUNCTIONS. 129 By distributing n in different ways between a, 6, c, . . . , we can obtain a large number of classes of functions. For example, if n = 5, we may take a = 5 ; -: = 1, ," = 1 ; = 10; • a = 3, 6 = 1, c = 1 ; e = 1, ,» = 20; S 2J S i1 • • • S r '■) G x cr,; ff if S :' T 2- • S 3' T ?> • • • S r t). If 9 130 THEORY OF SUBSTITUTIONS. then p < u, and if the group G x of

<^ — — ~ — - , and if the first line of the table does not contain any transposition, then some other line con- tains at least two. If these have one element in common, as (x a xp), (x a x y ), then, as we have seen in A), their product (x a xpx y ) occurs in 6r, . If they have no element in common, as (x a xp), (x y x&), then their product (x a Xp) (x y x s ) also occurs in G x . In either case (2, therefore contains a substitution of not more than four elements. C) There are (n — 1) (n — 2) substitutions of the form (x&aXp), {a-f,3 = 2, 3, . . . n). If therefore /' < (n— 1) (n — 2), and if G x con- tains no substitution of this form, some other line of the table con- tains at least two of them. A combination of these shows that G x contains substitutions which affect three, four, or five elements. Proceeding in this way, we obtain a series of results, certain of which we present here in the following Theorem I. 1) If the number /> of the values of a function is not greater than n — 1, the group of the function contains a sub- stitution of, at the most, three elements, including anrj prescribed Ttift 1 ) element. 2) If p is not greater than — — ^ — -, the group of the function contains a substitution of, at the most, four elements. 3) n(n — l)(n — 2) If p is „of greater than — ^ , the group of the function THE NUMBER OF VALUES OF INTEGRAL FUNCTIONS. 131 contains a substitution of, at the most, six elements. 4) If p is not . .. n(n— 1)(»— 2)...(w— fc + 1) ,, -.. . greater than — — i -, the group of the func- tion contains a substitution of, at the most, 2k elements. 5) If p is not greater than (n — 1) (w — 2) . . . (n — k-\-l), the group of the function contains a substitution of, at the most, 2k — 1 elements, including any prescribed element, so that the group contains at least n 2k— 1 such substitutions. By the aid of these results the question of the number of values of functions is reduced to that of the existence of groups contain- ing substitutions with a certain minimum number of elements. § 114. In combination with earlier theorems, the first of the results above leads to an important conclusion. From Chapter IV, Theorem I, we know that the order of an intransitive group is at the most (n — 1)!. Consequently, the num- ber of values of a function with an intransitive group is at least n ' -. '-zrr = n. For such a function therefore p cannot be less than n. (n — 1)! Again, the order of a non-primitive group is, at the most, 2! I — ! I , so that the number of values of a function with a non-primitive n\ group is at least . For n = 4, this number is less than n; *• 2 ' 2 but for n > 4, -it is greater than n. For such a function then, if n > 4, /> cannot be less than n. Again for the primitive groups it follows from Chapter IV, Theorem XVIII, in combination with the first result of Theorem I, § 113, that if p < n, the corresponding group is either alternating or symmetric, that is, p = 2 or 1. The non-primitive group for which n = 4, p = 4, r = 8 is already known to us, (§ 46). We have then Theorem II. If the number p of the values of a functioii is less than n, then either p = 1 or p = 2, and the group of the func- tion is either symmetric or alternating. An exception occurs only for n = 4, p = 3, r = 8, the corresponding group being that belong- ing to X x X.,-\- X-zXi. 132 THEORY OF SUBSTITUTIONS. £ 115. On account of the importance of the last theorem we add another proof based on different grounds. * Suppose (n — 1) ! substitutions of G } therefore rearrange only the p — 1 values

(n — 1 ) ! substitutions of G x there must be at least two, a and r, which produce the same rearrangement of cr.,, $p 3 , . . .

4 there is no such substitution (Chapter III. Theorem XIII). Consequently p > a. § 1 16. Passing to the more general question of the determina- tion of all functions whose number of values does not exceed a given limit dependent on n, we can dispose once for all of the less impor- tant cases of the intransitive and the non primitive groups. For the purpose we have only to employ the results already obtained in Chapter IV. In the case of intransitive groups we have found for the maxi- mum orders: 1) r=(n 1)!. Symmetric group of n — 1 elements. /' = //. (n — 1 >! 2) r = . Alternating group of // 1 elements. /> = 2 n. 3) /■ = 2\(n — 2)!. Combination of the symmetric group of n — 2 n(n — 1) elements with that of the two remaining elements, p = - 9 . 4) r — (n 2)!. Either the combination of the alternating group of n — 2 elements with the symmetric group of the two remaining elements; or the symmetric group of n 2 elements, In both cases /, = n(» — 1). Etc. * L. Kronecker: Monatsber. . p. 211. THE NUMBER OF VALUES OF INTEGRAL FUNCTIONS. 133 For the non- primitive groups we have 1) ?* = 2!l -~-! I . Two systems of non-primitivity containing each a jj elements. The group is a combination of the symmetric groups of both systems with the two substitutions of the systems them- ?i ! selves, p = - - . For n = 4, 6, 8, . . . we have i> = 3, 10, 35, . . . •(f)" 2) r = 3 ! I -Q- ! I . Three systems of non-primitivity. The group is a combination of the symmetric groups of the three systems with the 3 ! substitutions of the svstems themselves, p = — - — . For 8l(-lY n = 6, 9, 12, ... we have p = 15, 280, 5770, ... 137 3) r = 3 1 -=-! I . As in 2), except that only the alternating group of the three systems is employed, p = — — — ' For n = 6, 9, 12,. . . we have P = 30, 560, 11540, .. . 3 V3~ ! J The values of p increase, as is seen, with great rapidity. § 117. In extension of the results of § 113 we proceed now to examine the primitive groups which contain substitutions of four, but none of two or of three elements. Such a group G must contain substitutions of one of the two types The presence of s 2 requires that of s. 2 2 = (x a x c ) (x b x d ), which belongs to the former type. Disregarding the particular order in which the elements are numbered, we may therefore assume that the substitu- tion occurs in the group G. We transform s 5 with respect to all the substitutions of G and obtain in this way a series of substitutions of the same type which connect x l , x 2 , x s , a- 4 with all the remaining elements (Chapter IV, Theorem XIX). The group G therefore includes substitutions 134 THEORY OF SUBSTITUTIONS. similar to s 5 which contain besides some of the old elements .c, . .r t other new elements o? 6 , x t , .» ; , . . . This can happen in three different ways, according as one, two, or three of the old elements are retained. Noting again that it is only the nature of the connection of the old elements with the new, not the order of designation of the elements that is of importance, we recognize that there are only five typical cases : \X\Xl) \X%X h ), Kp^V^z) V^2^i)i (XiX 5 ) (X 2 X 6 ), (^l-^s) \ X 3 X 6)j \X\X 5 ) [X & Xi). In the first case, for example, it is indifferent whether we take {XyXo) (X^^j, yX^X.)) {XiX^}) (X^X^ {XyX^J, [X^X}) \X 2 X 5 ) ', and in the last we may replace a^ by x 2 , x 3 , or x t , etc. The first and fifth cases are to be rejected, since their presence is at once found to be inconsistent with the assumed character of the group. Thus we have \X\X 2 j \X 3 Xi) • \X^X 2 j [X^X^J — \X2,X i X^f i \\X1X2) {X^X^ • (XiX 5 ) {X 6 Xi) J = {X l X 5 X 2 ), the resulting substitutions in each case being inadmissible. There remain therefore only three cases to be examined, accord- ing as G contains, beside s, , one or the other of the substitutions JL) yX^Xz) (X2X5)) B) {x,x^ (x 2 x 6 ), the first case involving one new element, the last two cases two new elements each. § 118. A) The primitive group G contains the substitutions n- = yx i X 2 ) {X^X^), S 4 = (X^s) (X 2 ;l' : , ). and consequently also t ^ S5S4 == \XiX 5 X2X s x i ), Sj ^ taj — yx 2 x$) yp^i'^a)' Since t is a circular substitution of prime order 5, it follows from § 83, Corollary I, that if n^l, is at least three-fold transitive. Then G must contain a substitution u, which does not affect x t but THE NUMBER OF VALUES OF INTEGRAL FUNCTIONS. 135 replaces ar, by x 6 and x 3 by x 1 . If we transform s r , with respect to this substitution, we obtain s' = u~\u = (x^) (x 1 x a ). If x„ is contained among j» 2 ,aj 8 ,aJ 4 , oj 5 , then .s' and 8, have only one element in common and if x a is contained among x $ ,x 9 ,... then s' and s 5 have only one element in common. Both alternatives therefore lead to the rejected fifth case of the preceding Section. If n>_l, G becomes either the alternating or the symmetric group. There is in this case no group of the required kind. For n = 4 it is readily seen that there are two types of groups with substitutions of not less than four elements, both of which are however non- primitive. Groups of the type A) therefore occur only for n — 5 or n — 6. For n = 5 we have first the group of order 10, If we add to G 1 the substitution a = (x^x^), we obtain a second group of order 20 The latter group is that given on p. 39. 6r, and Gr 2 exhaust all the types for n = 5. For B = 6we obtain a group G t of the required type by adding, to G 2 the substitution (x^) (x 2 x 3 ). Since G x is of order 10, the transitive group 6r 4 must be at least of order 60 (Theorem II, Chapter IV). And again, since {x x x^} (x 2 x 3 ) 6 ) (x,» :i ). These three substitutions are not sufficient to connect the six ele- ments .«•, , x 2 , . . . x transitively, there being no connection between x 3 , x 4 and .»•,, .»•_,, .«•-,,.«•,,. The group must therefore (§ 83) contain another substitution of the type (x a xp) (x y x & ) which connects Xi i x 2 ,x 6 ,x i with other elements. If this substitution should con- tain three of the elements ,r, , x., , « 5 , x 6 and only one new one, it would have three elements in common with v. This would lead either to to the type A) or to the rejected first case of § 117. If the new substitutions contained only one of the new elements a?i, x 3 , x b , a? B and three new ones, then we should have the fifth case of § 117, and this is also to be rejected. There remains only the case where the new substitution connects two of the elements ,r,, x 2 , x 5 , x t , with two others. It must then be of one of the forms \X\X a ) [X 2 Xi,), y^'v^a) [p^V^bJl [P^V^a) V^V^fijj \x 2 x a ) [x s Xf,)f {x 2 x„) {x^X/,), (x r ,x ri ) {x 6 Xi,). Of these the first, third, fourth and sixth stand in the relation defined by C) to r, while the first, second, fifth and sixth stand in the same relation to v. All the groups B) therefore occur under either A) or C), and we may pass at once to the last case. § 120. C). In this case the required group contains tf-j = (x x X. 2 ) fax^) <7. 2 = fax 5 ) fax 6 ), <7 3 = *{■ l T. 2 ff 1 = (x 2 x 5 ) (x t x 6 ). We consider first the case n = 6. The elements x l ,x 2 ,x- 1 are not yet connected with x 3 ,x Ai x 6 . There must be a connecting substitution in the group of the type (x a xp) (x y x&), where we may assume that x a is contained among the the three elements x u x 2 ,x b . If x a were x 2 or x 61 then we should obtain, by transformation with respect to t, or — \XiX 3 ) \X 2 X a ), and since a x has three elements in common with _9, at least 4-fold transitive. G contains therefore the sub- stitutions T = (CCj) {X 2 X 3 ) (a?4#*5 . . . ) , r tfjT =. (X]a" 3 ) (x 2 x 5 ), so that we return in every case to the type A). For n >$ there is therefore no group of the required type. Theorem III. // the degree of a group, which contains substitutions of four, but none of three or of two elements, exceeds 8, the group is either intransitive or non-primitive. Combining this result with those of § 113 and § 116, we have Theorem IV. If the number p of the values of a function is not greater than %n(n — 1), then if n > 8, either 1) p = %n(n — 1), and the function is symmetric in n — 2 elements on the one hand and in the two remaining elements on the other, or 2) f = 2n, and the function is alternating in n — 1 elements, or 3) p = n, and the function is symmetric in n — 1 elements, or 4) p = 1 or 2, and the function is symmetric or alternating in all the n elements* § 122. We insert here a lemma which we shall need in the proof of a more general theorem, f From § 83, Corollary II, a primitive group, which does not include the alternating group, cannot contain a circular substitution *Cauchy:Journ. rte PEcole Polytech. X Cahier; Bertrand: Ibid. XXX Cahier; Abel: Oeuvres completes I, pp. 13-21; J. A. Serret: Journ. del'Ecole Polytecb. XXXII Cahier; C.Jordan: Traiteetc, pp. 07-75. tC. Jordan: Traite etc.. p. 664. Note C. THE NUMBER OF VALUES OF INTEGRAL FUNCTIONS. 139 2n of a prime degree less than ■=- . If p is any prime number less In than -=- , and if p ■' is the highest power of p which is contained in o n ! , then the order of a primitive group G is not divisible by p f . For otherwise G would contain a subgroup which would be similar to the group K of degree n and order p f (§ 39). But the latter group by construction contains a circular substitution of degree p, and the same must therefore be true of G. Consequently p = — must contain the factor p at least once. What has been proven for p is true of any prime number less than -=- and consequently for their product. We have then o Theorem V. If the group of a function ivith more than tivo values is primitive, the number of values of the function is a multiple of the product of all the prime numbers ivhich are less than -K-. o § 123. By the aid of this result we can prove the following Theorem VI. If k is any constant number, a function of n elements ivhich is symmetric or alternating with respect to n — k of them has fewer values than those functions which have not this property. For small values of n exceptions occur, but if n exceeds a certain limit dependent on k, the theorem is rigidly true.* If

n — k i. e., n> 2& + 1. The maximum order of the group is consequently (»— k— 1)! (fc + ll, and the minimum number of values of 4> is n\ _n(n — 1) (n — 2) . . . (n — k) (n—k—iy. (A; + l!) _ 1-2-3. . ..(&+1) \ It appears at once that the minimum B) exceeds the maximum A), as soon as »>fc + 2(fc+l)! This is therefore the limit above which, in the first case, the theo- orem admits of no exception. § 125. In the second case 4> is transitive in n — /. elements (x>_k), but it is neither alternating nor symmetric in these ele- ments. The group G of ff 2 T 2> ff 3 r 8> • • • ff a~a, • • ■ ^Pi • ■ • where, however, one and the same '2A\ Consequently G contains substitutions "V^? ffprp, in which - a — ~p but is n\ _ (n — x)\ n(n — 1) . . . (n — *-f-l) ldR(n — x)~ R(n — x)' ~TT We have now still to determine R(n — x), the maximum order of (n — z )! a non- alternating transitive group of » — x elements, or -— — — , the minimum number of values of a non-alternating transitive function of n — /. elements. If this function is non-primitive in the n — * elements, it follows that the minimum number of values is n . / M (n— *)(n— *— l)...(^P"+l) C,) (n — x)\ _ v V Z J 2 j [i(n— «)]!J» ~ a " "[T^ 7 *)] ! Substituting this value in C) we obtain for the minimum number of values of 4' n(n— 1 )...(/< — /+ 1)(>/— x) ... (^=-^-+1 ) C ) i ■ 142 THEORY OF SUBSTITUTIONS. We compare this number with the maximum number A) and examine whether, above a certain limit for n, C\) becomes greater than A), i. e., whether n (n-l)...(»-* + l)(n-x)...(^+l) >4*! ^-^\ n(n— 1). . .(n — x + 1). For sufficiently large n we have I — ~ |- 1 I < n — k-\-\. We have therefore to prove that (n-k)(n-k-l)...(p^+l) >4z!— -— ! This is shown at once, if we write the right hand member in the form For the first factor is constant as n increases, and the ratio of the left hand member to the second parenthesis has for its limit n-\-K 2-"—*. § 127. Finally, if the function 4' of the n — x elements is prim- itive, we recur to the lemma of § 122. From this it follows that the minimum number of values of 2xl(n _ z) (w __ x _ 1} (n—jb + l). The right hand member of this inequality will be greatly increased if we replace every n — /■ — a by the first factor n — *. There are k — /■ factors of the form n — * — o. These will be replaced by THE NUMBER OF VALUES OF INTEGRAL FUNCTIONS. 143 (n — x) k ~ K . If we write then v = ^ — -, we have only to prove that for sufficiently large v, P(")>[2(|)*-^!]^-«, or P(y) „*-« >[ 2 (*)*-«*!]. This can be shown inductively by actual calculation, or by the employment of the theorem of Tchebichef, that if v > 3, there is always a prime number betiveen v and 2 v — 2. For we have from this theorem P(2v)>vP(v), (v)*-« = 2*-*»/ fc -« P(2v) > P(v) V (2^)*-* «*-« 2 A '-«' Now whatever value the first quotient on the right may have, we can always take t so great that the left hand member of P(2',) ^ P(v) (2V)*-« > PW /^V >*-« V.2 k - K J increases without limit, if only v is taken greater than 2 k '~ K . The proof of the theorem is now complete. The limits here obtained are obviously far too high. In every special case it is possible to diminish them. As we have, however, already treated the special cases as far as p = %n(n — 1), it does not seem necessary, from the present point of view, J;o carry these inves- tigations further. CHAPTER VII. CERTAIN SPECIAL CLASSES OF GROUPS. £ 1 28. We recur now to the results obtained in § 48, and deduce from these certain further important conclusions.* Suppose that a group G is of order r =p a m, where p is a prime number and m is prime to p. We have seen that G contains a sub- group H of order p a . Let J be the greatest subgroup of G which is commutative with H. J contains H, and the order of J is there- fore p a i, where i is a divisor of m and is consequently prime to p. Excepting the substitutions of H, J contains no substitution of an order p$. For if such a substitution were present, its powers would form a group L of order pP. But if in A) of § 48 we take for Gr, , H 1 , .ST] the present groups J, L, K, then since n= '/, '/, ""' and for the same reason as before d y — p a in at least one case, and therefore M — u y x H a i substitutions of J, it follows that there are always exactly pH substitutions of G which transform H KYI into any one of its conjugates. There are therefore — of the latter. *L.8ylow: Math. Ann. V. 684-94. CERTAIN SPECIAL CLASSES OF GROUPS. 145 Finally, if we replace G, H x , K x of A) § 48 by G, J, H, we have /■ _ p a m _ p a i p°-i Since if, is contained in J, , we must have d, = p a , and since J con- tains no other substitutions of order p&, no other d can be equal to p a . It follows that r = p a i (kp -j- 1), m = i(kp -\- 1) . The group H has therefore kp -{- 1 conjugates with respect to G. We have then the following Theorem I. If the order r of a group G is divisible by p a but by no higher power of the prime number p, and if H is one of the subgroups of order p a contained in G, and J of order pH the largest subgroup of G which is commutative icith H, then the order of G is r=pH(kp-{-l). Every subgroup of order p a contained in G is conjugate to H. Of these conjugate groups there are kp -\- 1, and every one of them can be obtained from H by pH different transformations. § 1*29. In the discussion of isomorphism we have met with tran- sitive groups whose degree and order are equal. In the following Sections we shall designate such groups as the groups i-\ If we regard all simply isomorphic transitive groups, for which therefore the orders r are all equal, as forming a class, then every such class contains one and only one type of a group i-' (§ 98). The construction of all the groups ii of degree and order /• therefore furnishes representatives of all the classes belonging to /•, together with the number of these classes. The construction of these typi- cal groups is of especial importance, because isomorphic groups have the same factors of composition, and the latter play an impor- tant part in the algebraic solution of equations. One type can be established at once, in its full generality. This type is formed by the powers of a circular substitution. A group Li of this type is called a cyclical group, and every function of n ele- ments which belongs to a cyclical group is called a cyclical function. 10 140 THEORY OF SUBSTITUTIONS. We limit ourselves to the consideration of cyclical groups of prime degree p. If s = (a?, x 2 . . . x p ), and if w is any primitive p th root of unity, then V = (•'*! + w #2 + W "V ( + • • ■ + <"'' l x p ) p is a cyclical function belonging to the group G = [1, s, s 2 , . . . s p ~ 1 '\. For X 2 + . . . + a>*- l Xp )* =

0. These sub- groups would have only the identical substitution in common. They would therefore contain in all (p- L)(p*+l) + l=p[(p -l)* + l]>pg substitutions. This being impossible, we must have x — 0. CERTAIN SPECIAL CLASSES OF GROUPS. 147 The subgroup H coutains only p substitutions ; the rest are all of order q. Their number is pq—p=(q—l)p. There are therefore p subgroups of order q, and consequently from Theorem I we must have p — 1 p = *q + l, / q ' that is, q must be a divisor of p — 1. Only in this case can there be any new type Q. 3) The group if is a self -conjugate subgroup of fi. Conse- quently every substitution t of order q must transform the substitu- tion s of H into s°, where a might also be equal to 1. We write (where the upper indices are merely indices, not exponents). Then no cycle of t can contain two elements with the same upper index. For otherwise in some power of t one of these elements would follow the other, and if this power of t were multiplied by a proper power of s, one of the elements would be removed, With a proper choice of notation, we may therefore take for one cycle of t {x^XiX* . . . xf). It follows then from t- l st = s n that t replaces x.!' by x a+l h+i , x-f by x ia+1 b+1 , . . . x a+1 h by a* arr + ] 6 + 1 , ... so that we have t == \Xi X} . . . X 1 ) . . . \X a -)_ i ) that q is a divisor of p — 1, as we have already shown; further that a, belong- ing to the exponent q, has q — 1 values a,, a,, . . . a rj ,; finally that all these values are congruent (mod. p) to the powers of any one among them. From t \s t = s" follows *- 2 sf'- = s« 2 , t~ 3 sf = s« :i , ... so that, if 8 is transformed by t into any one of the powers .s a A, there are also substitutions in Q which transform s into .s"i, s% . . . «*«— i. Accordingly the particular choice of r/ A has no influence on the resulting group, so that if there is any type ii generated by substi- tutions s and t, there is only one. The group formed by the powers of t being commutative with that formed by the powers of s, the combination of these two sub- stitutions gives rise to a group exactly of order pq. The remaining pq — p — q-\- 1 substitutions of the group are the firsts/ — 1 powers of thep — 1 substitutions conjugate to t 8 -p+i t8 p-i = (xjxffxf? . . . xtf) ... (/5 = 2, 3, . . .p). If ji and ij are unequal, we have therefore only one new type J-\ § 131. Finally we determine all types of groups Q of degree and order p 2 . 1) The cyclical type, characterized by the presence of a substi- tution of order p", is already known. 2) If there are other types, none of them can contain a sub- stitution of order p\ There are therefore in every case p l — 1 sub- stitutions of order p and one of order 1. If s is any substitution of --. and t any other, not a power of s, then ii is fully determined by 8 and t. For all the products s"t" (a,b = 0,l,2,...p—l) arp different, and therefore fl = (>•*»] (0,6 = 0,1,2, ....p— 1). We must have therefore t.s — .s 6 i t<\ , I 'a = s a -i e «, . . . l p l 8 = & ~ i & - 1 . If now two of the exponents d are equal, it follows from CERTAIN SPECIAL CLASSES OF GROUPS. 149 t a s = s s t% t''s = s s t" (a = b, e=fe') that {t"s)-\t"s) = s Hs = (sH^-^t") = P. Since for t we may write t, it therefore appears that Q contains a substitution t which is transformed by 8 into one of its powers V. The same result holds, if all the exponents d are different. For one of them is then equal to 1, since none of them can be 0, and from t" s = st € follows s~ l t"s= t*. 3) There is therefore always a substitution / I rg% 1 /y* * /V» 1"\ //y» ~/y * /y» -\ //y» P /y P /y* P \ V 1 tA. j «A'2 • * • **" 1> / V I 2 * * * f) ) ' • • V **■' 1 **-'2 • • • ** / « / which is transformed by s into a power of itself V*. As in the pre ceding Section, we may take for one cycle of s ( ™ !™ 2 rp P\ y«A/] tA. j . . . tA j y. Then from s~ 1 ts = f' follows If the second cycle is to close after exactly p elements, we must have a 1 ' + 1 = 2, a p =l (modp). This is possible only if a = 1. Accordingly / /-y. * ^yi - /y» P\ ( ry* * /y» * *yi _P \ / /y» ^ -y» " J' | — ^iA j cA/j . . . u. j ^ \«*-2 ** / 2 ' • • **'2 / • • • V**i) "•> • • • p )• Thep + 1 substitutions Oa ' . Oka M a « • • Ol are all different and no one of them is a power of any other one. Their first p — 1 powers together with the identical substitution form the group £. Summarizing the preceding results we have Theorem II. There are three types of groups Q, for which the degree and order are equal to the product of tivo prime num- bers : 1) The cyclical type, 2) one type of order pq (p > q), 3) one type of order p 2 . The first and third types are always pres- ent; the second occurs only when q is a divisor of p—1. § 132. We consider now another category of groups, character- ized by the property that their substitutions leave no element, or 150 THEORY OF SUBSTITUTIONS. only one element, or all the elements unchanged. The degree of the groups we assume to be a prime number p. Every substitution of such a group is regular, i. e., is composed of equal cycles. For otherwise in a proper power of the substitu- tion, different from the identity, two or more of the elements would be removed. The substitutions which affect all the elements are cyclical, for p is a prime number. From this it follows that the groups are tran- sitive, and again, from Theorem IX, Chapter IV, that the number of substitutions which affect all the elements is p — 1. We may therefore assume that S — (XiX-20C^ . . . Xp) and its first p — 1 powers are the only substitutions of p elements which occur in the required group. The problem then reduces to the determination of those substi- tutions which affect exactly p — 1 elements. If t is any one of these, then t~ l st, being similar to s, and therefore affecting all the ele- ments, must be a power of s t St — S — ^#*jfl?j ^. m X t _|_ ■>„, . . . j, where every index is to be replaced by its least positive remainder (mod 2?). Since it is merely a matter of notation which element is not affected by t, we may assume that .r, is the unaffected element. It follows that t -— [XoX m _j_ j X„,2 _|_ j X „,8 _j_ j ...)... ^ X„ _f_ ! X a ,„ _|_ i X n ,„2 _|_ i . . . ) . . . If now — 1 elements each. For every cycle of ^ CERTAIN SPECIAL CLASSES OF GROUPS. 151 (mod. p) am ZJ r l==a + lj m~= gP'^l, p-1 and this first happens when z = If there is any further substitution t v which leaves x r unchanged and which replaces every # + , by x ag v +11 then t^tf replaces every x a + l by Xagiii+Pv+i. If now we take « and /3 so that afi-\-fiv is congruent (mod. p) to the smallest common divisor at of // and v, we have in f — f af ■'(O — ■'Jl ^V a substitution of the group, of which both t^ and t v are powers. Proceeding in this way, we can express all the substitutions which leave ac 3 unchanged as powers of a single one among them t a , where g° is the lowest power of g to which a substitution t of the group corresponds. p — 1 The group is determined by s and t a . Since t a is of order , it follows from Theorem II, Chapter IV, that the group contains in a U PSJl L substitutions, a may be taken arbitrarily among the divisors of p — 1. § 133. To obtain a function belonging to the group just con- sidered, we start with the cyclical function belonging to s <\ = (», + "a* + " 2 *3 + • • ■ + w*- 1 *,)*, where w is any primitive p th root of unity. Applying to 0, the successive powers of t a , we obtain 02 = (*i+ a, a*,«r +I + w 2 aJ 2j , «r + 1 )*, The powers of s, forming a self-conjugate subgroup of the given group, leave all the 0's unchanged. The powers of t„, and conse- quently all the substitutions s a t a b of the group, merely permute the 0's among themselves. Every symmetric function of 01,02, ...0,-i 152 THEORY OF SUBSTITUTIONS. is therefore unchanged by every substitution of the group. Ac- cordingly if is any arbitrary quantity, the function Y > '\ the relation az-\- [1 a x z-\- {i yz + 8 y t z + 8 (mod. p) 154 THEORY OF SUBSTITUTIONS. were possible, it would follow from the comparison of the coeffi- cients of z 2 , z\ and z°, with the aid of D', that if a, a', ,;, ,5', . . . are real, «../ 3 __r_«J_ / ad— fly _ ~ZT = -oT = -— t = tt- - A/-Tv 77-7 = ^±l ( mod. P)- If, therefore, we restrict the range of the values of a, /?, ^, '5 to 0,1,2, . . . p — 1, there are always two and only two different sys- tems of coefficients which give the same substitution u. With D') it is assumed that ad — ,iy is different from 0. This restriction is necessary, for the symbol u can represent a substitu- tion only if different initial values of z give rise to different final values of z, i. e., if the congruence az-\-[i aZj -j-i9 yz+8 yz, + Z (mod. p) is impossible. This is ensured by the assumption ad — i3y==0. We determine now how many elements are unchanged by the substitution u. An index z can only remain unchanged by u if E) r z 2 + ('> — ")z — 0=0 (mod. p). There are accordingly four distinct cases: a) The two roots of E) are imaginary. This happens if m T 1 {ad— p r =±l) is a quadratic non-remainder (mod. p). The corresponding substi- tutions affect all the elements ac 0J .<•, , x 2 , . . . x p _ 1} x^ . b) The two roots of E) coincide. This happens if (nrO* 1 - ^ mod -^) («*— /»r=±l). The corresponding substitutions leave one element unchanged. c) The two roots of E) are real and distinct. This happens if (=£)'* i (.»-/»/= ±i) is a quadratic remainder (mod. p). The corresponding substitutions leave two elements unchanged. d) The equation E) may vanish identically. This happens if r=0, ,5 = 0, a=d (mod. p). CERTAIN SPECIAL CLASSES OF GROUPS. 155 The corresponding substitution leaves all the elements unchanged. Finally we observe also that az + fi yZ + d z ' (r«i+^i)*+(rft+ M i) M) yz + d y x z + <\ N) (ad—Py) («!'?! — yJ, r ,) = (aaj + p ri ) (yPj + 38 r ) — iafi } + fid 1 )( r a l + 9 r i). We proceed now to collect our results. If we take a not E (mod. p), and /3 and /-arbitrarily, then for each of the (p — l)p 2 resulting systems we obtain two solutions of D'). Since however there are always two systems of coefficients which give the same substitutions u, we have in all, in the present case, p' — p 2 substitu- tions. Again, if we take a =0 (mod. p) and 8 arbitrarily, then restricting /5 to the values 1, 2, . ..p — 1, we obtain from D') for every system a, S, (3 two values of y; but as two systems of coef- ficients give the same u, we have in this case p{p — 1) substitutions. There are therefore in all p 3 — P = (p-\-l) p(p — 1) fractional linear substitutions (mod. p). From M) it appears that these form 1 1 ' ~\~ 1 ) P ( P — 1 ) a group. Among them there are - — % - substitutions a which correspond to the upper sign in D'). From M) and N) it is clear that these also form a group. This latter group is called "the group of the modular equations for p".* Both groups contain only substitutions which affect either p -\- 1, or p, or p — 1 elements, or no element. Those substitutions which leave the element x n unchanged, for which accordingly y=0, form the metacyclic group of § 134. As the latter is two- fold transitive, it follows (Theorem XIII, Chapter IV) that the group of order (p -\- l)p(p — 1) is three-fold transitive. Theorem III. The fractional linear substitutions (mod. p) form a group of degree p-\-l and of order (p -|- 1) p (p — 1). Those of which the determinants are quadratic remainders (mod.p) ( p -\- 1) jo ( p — 1) foi-m a subgroupt of order ^— £ '■> the group of the modu- lar equations for p. If any substitution of these groups leaves more than tivo elements unchanged, it reduces to identity. The first of the two groups is three fold transitive. * Cf. J. Gierster: Math. Ann. XVIII, p. 319. 150 THEORY OF SUBSTITUTIONS. To construct a function belonging to the group of the fractional linear substitutions, we form first as in § 183, a function '/', of the elements x , .r,, .<•., . . . x p , which belongs to the group of substi- tutions t=\z ■■ ,3z+ a\ (mod. p). (« = 0, 1, 2, . . . /- -1; /3=1, 2, . . .p— 1) The substitutions u, applied to '/'", , produce p -\- 1 values ' 1 > ' 2 J • ■ • r p + 1 > which these substitutions merely permute among themselves. Ac cordingly, if '/'' is any undetermined quantity, the function 2 = (♦/•— '/■•,)('/■•— '/';)...<'/ — v; + 1 ) belongs to the given group. § 187. We have now finally to turn our attention to those groups all the substitutions of which are commutative. We employ here a general method of treatment of very exten- sive application.* Suppose that 0', 6", (>'" . . . are a series of elements of finite number, and of such a nature that from any two of them a third one can be obtained by means of a certain definite process. If the result of this process is indicated by /, there is to be, then, for every two elements 0', 0", which may also coincide, a third element 9'", such that /("' (>") = <>'". We will suppose further that /(*', 0") =f(0", 6% flo',f(o",o'")]=f[f(o',o" } ji'"i i , but that, if 0" and 0'" are different from each other, then These assumptions having been made, the operation indicated by / possesses the associative and commutative property of ordinary multiplication, and we may accordingly replace the symbol /("', "") by the product 0'0", if in the place of complete equality we employ the idea of equivalence. Indicating the latter relation by the usual sign oo , the equivalence II' n" co ""■' is, then, defined by the equation f(0\ II" ) = ()'". *\j. Kronecker: Monatsber. d. Berl. Akad.. 1870. p. 881. The following is taken for the mosl part verbatim from this article. CERTAIN SPECIAL CLASSES OF GROUPS. 157 Since the number of the elements 0, which we will denote by n, is assumed to be Unite, these elements have the following properties: I) Among the various powers of an element there are always some which are equivalent to unity. The exponents of all these powers are integral multiples of one among them, to which may be said .to belong. II) If any belongs to an exponent v, then there are elements belonging to every divisor of v . III) If the exponents p and ' ■» ',' : coii" We retain the sign of equivalence to indicate the original more lim- ited relation. If now we select from the elements any complete system of elements which are not relatively equivalent to one another, this subordinate system satisfies all the conditions imposed on the entire system and therefore possesses all the properties enumerated above. In particular there will be a number ji., , corresponding to w. ls such that the u., th power of every of the new system is relatively equiv- alent to unity, i. e., ^"-c\o"/. Again there are elements (>,, in the new system of which no power lower than the >i, th is relatively equivalent to unity. Since the equivalence (>"'co 1 holds for every element, and consequently a fortiori every i'"- is relatively equiva- 158 THEORY OF SUBSTITUTIONS. lent to unity, it follows from I) than a, is equal to n 2 or is a multiple of )i . If now and if both sides are raised to the power — , we obtain, writing — = m. the equivalence (\ icvdI. From this it follows that, since 0j belongs to the exponent n l , m is an integer and k is therefore a multiple of n., . There is therefore an element (>.,, defined by the equivalence ii,n;"coi> n or a oo0„8 1 n i- m of which the u., th power is not only relatively equivalent, but also absolutely equivalent to unity. This element belongs both rela- tively and absolutely, to the exponent it,, for we have the relation ".,"■- co 0„ ""- ",">"•- '" "'-co ('V' 7 '," 1 "-"'" "-co "-co 1. Proceeding further, if we now regard any two elements 0' and 0" as relatively equivalent when 0' 0* 0* co 0", we obtain, corresponding to (>.,, an element 8 belonging to the expo- nent n 3 , where n z is equal to n., or a divisor of n 2 ; and so on. We obtain therefore in this way a fundamental system of elements 11 \ > ".> > ".>. 5 • • • which has the property that the expressions 0,*i0 2 *20 8 V . . (h,= 1,2,.. .n) include in the sense of equivalence every element once and only once. The number w, , n., , w 3 , . . . , to which the elements mi , <>., , d t , . . belong, are such that every one of them is equal to or is a multiple of the next following. The product », n., n^ ... is equal to the entire number n of the elements 0, and this number n accordingly contains no other prime factors than those which occur in the first number ?i, . § 1 89. In the present case the elements are to be replaced by substitutions every two of which are commutative. The number n of the elements becomes the order 7* of the group. We have then CERTAIN SPECIAL CLASSES OF GROUPS. 159 Theorem IV. // all the substitutions of a group are com- mutative, there is a fundamental system of substitutions s, , s 2 , s 3 , . . . which possesses the property that the products V WV:: . . • (hi =1,2,... ?-,) include every substitution of the group once and only once. The numbers r lt r. 2 , r 8 , . . . are the orders of s u s 2 , s 3 , . . . and are such that every one is equal to or is divisible by the next following. The product of these orders r,, r 2 , r 3 , . . . is equal to the order r of the group. The number i\ is determined as the maximum of the orders of the several substitutions. On the other hand the corresponding substi- tution s x is not fully determined, but may be replaced by any other substitution s/ of order r, . According then as we start from s l or s/, the values of r 2 ,r 8 ,.;. might be different. We shall now show that this is not the case. In the first place it is plain that if several successive s's belong to the same exponent r, these s's may be permuted among them- selves, without any change in the r's. Moreover, every s a can be replaced by s a f l s a+i v s a+2 T • • • without any change in the r's, pro- vided only that fi is prime to r a . If now the given group can be expressed in the two different forms * W • • • (h, = 1, 2, . . . r,), (w) shall all be different and shall be identical, apart from their order, with the system 1, 2, 3, . . . n. On the other hand it is readily shown that every substitution can be expressed in this notation.. For if it is required that ?(1) = /,. 90(2) = ^, . . . c{h) - / . we can construct, by the aid of Lagrange's interpolation formula from F$ = (z— l)(z— 2). . .(z- n) a function ' l — 1= 1. For » == p, the functions (i)]";+ • • • +[Kp— i]"' =(p—i)A p _ 1 (. m y = — A t , ,'"" (mod. p). If now (0)]|>— ?(1)] . . . — ] .,-z" — «z- ; ? = (1 — a)z — ,3 (mod. p). *Hermite: Comptes reiulustle I'Academie cles Sciences, vr. 11 162 THEORY OF SUBSTITUTIONS. Accordingly, if a=pO, the linear congruence (1 — a)z — /3=(mod. p) is satisfied by the p integers .) + b A (mod. m) c A {i n z.,, . . . g* + l) = ?A(zi,*s, • • • 2*) + Ca (mod. ///). From these congruences it appears at once that the c' A 's are linear functions of the z\b, having for their constant terms tf A = /x(0,0,0, . . . 0). The remaining coefficients are then readily found. In fact, we have c- A (~, , z 2 , ...z k ) = a K i, + b x z 2 -f . . . + e K z,, + 5 A , and therefore t = : . : | . . . . z k «,2, + b t Z 2 + . . . + C,«, + '', . a 2 z, + b,z., +....+ c 2 «* + ''.,... . Conversely all substitutions of this type transform the group G into itself. Thus, for example, t transforms s, „ into a -., \-biZ 2 -\- . . . +c 1 z k + 8 11 . . . a,( s, + 1 ) + &,2r 2 + ...+,., ?,..+ *, i. e., into the arithmetic substitution 'H -.'•■•■ -/. »1 I a i ) "J I a 2j • • • ** I "'■ ~ *aj . a. . . . . a/..- By left hand multiplication by • s '6, , 8, , . . . 8* ' = s 6, . &J ... 5,.. we can reduce t to the form f = \z u Z 2 ,...Z k d l Z 1 + b l Z 2 + ...+C X 2 . " .: j-/,.;.-j- . . . + <•.: . . . . a»g, -hM 2 + • • • +c*»*|. Such a substitution is called a geometric substitution.* We proceed to examiue this type. We have already demon- strated Theorem II. All geometric substitutions and their combi- nations with the arithmetic substitutions, and no others, arc com- mutative with the group of the arithmetic substitutions. •Cauchy: loc. clt. ANALYTICAL REPRESENTATION OF SUBSTITUTIONS. 165 § 145. We have first of all to determine whether the con- stants a A , fr A , . . . c x can be taken arbitrarily. They must certainly be subjected to one condition, since two elements ■*"-,, -., _ v and .» b] ^ gj. must not be converted into the same element unless the indices z t , z.,, . . . z k coincide in order with ',,',, . . ,~ k . More generally, given any system of indices X x , -,, . . . ~ k , it is necessary that from a i z l -\-b j z.,-\-...-\-c i z l ~i, a.,Z\ -\-b. 2 z. 2 -\- ... + c 2 2/. = T, , . . . (mod. m) the indices z n z.,, . . . z k shall be determined without ambiguity. In other words, the m k systems of values z must give rise to an equal number of systems of values ~ . The necessary and sufficient conditions for this is that the congruences «$, + b^., -j- . . . + c,2k=0, a. 2 z x -\- b,z., -\- . . . + Co^.^0, . . . (mod. m) shall admit only the one solution z x = 0, z 2 = 0, . . . z h = 0. If the determinant of the coefficients is denoted by J, these congruences are equivalent to J • Zi = 0, J . z. 2 = 0, . . . J -2,, = (mod. />/ ). The required condition is therefore satisfied if and only if J is prime to m. We have then Theorem III. In order that the symbol t = z x , z, ,...z k a x z x + b x z, + . . . + c t z k , a,z l + b,z, + . . . + c,z L , . . . | (mod m) may denote a (geometric) substitution, it is necessary and sufficient that a M 6 lf . . .c x a. 2 , b>, c, a*, b k , . . . C; should be prime to the modulus m. § 146. From this consideration it is now possible to determine the number r of the geometrical substitutions corresponding to a given modulus m. We denote the number of distinct systems of p integers which are less than m and prime to m by [m, p\. It is to be understood that any number of the ;> integers of a system may coincide. 106 THEORY OF SUBSTITUTIONS. Suppose N to be the number of those geometric substitutions 1, t.., t :i , . . . which leave the first index z x unchanged. If then r a is any substitution which replaces z, by a l z l -\- b^., -\- . . . +c,z*., "then r i) ^i r .< t» T n • • • are all the substitutions which produce this effect, and these are all different from one another. Similarly, if r .. replaces 2, by a, / «i + 6] / « a + . . . + C|'z fc , then r 8 , £ 2 t 8 ,£ 8 t 8 , . . . are all the substitutions which produce this effect, and these are all different, and so on. We obtain therefore the number r of all the possible geometric substitutions by multiplying N by the number of substi- tutions 1, t 2 , t 8j . . . The choice of the systems a n b lt . . . c,; a/, 6/, . . . c/; . . . is limited by the condition that that the integers of a system cannot have a same common factor with ra. There are therefore [ra, A:] such systems, and an equal number of substitutions 1, r 2 , t 8 , . . . Consequently r= [ra, k]N. The substitutions t are of the form |z,,z,, . . .z k z^a^ + boZ,-^ c,z k , . . . a& 1 + b k z 2 + . . . + c k z k \ . (mod.?//). Since a 2 ,a 3 , . . . a k do not occur in the expression of the discrimi- nant J, these integers can be chosen arbitrarily, i. e., in m ' dif- ferent ways. The b K , . . . c A are subject to the condition that & 2 , . . . C 8 lh, ■ ■ -c k must be prime to tn. If the number of systems here admissible is r, we have r = [ra, &] ra* - V. The number r' has the same significance for a substitution of k — 1 indices (mod. ra) as r for k indices. Consequently r = [ra, A;] ra fc_1 [ra, A- — 1] m k ~ V, and so on. We obtain therefore finally r = [>//,/, | ///' l [ra,ifc— l]ra* -". . . [ra, 2|/" ", where r (/ '' corresponds to a single index, and therefore r< fc_l ) ~[ra, 1]. Hence ANALYTICAL REPRESENTATION OF SUBSTITUTIONS. 167 4) r = [m, k I m h ~ l [m, A; — 1] m k ~ 2 . . . [ra, 2] ra [ra, 1] . The evaluation of [m, A-j presents little difficulty. We limit our- selves to the simple case where ra is a prime number p, this being the only case which we shall hereafter have occasion to employ. We have then evidently 5) [P»/°]=P P — li since only the combination 0, , ... is to be excluded. By the aid of 5), we obtain from 4) 6) r = (p k —l)p k - 1 (p k - 1 —l)p k - a . . . (p 2 — l)p(p—l) = (p A ' — 1) (p k — p) (p k — p~) . . . (p' c — p k ~ 1 ).* § 147. The entire system of the geometric substitutions (mod. m) forms a group the order of which is determined from 4) or from 6). This group is known as the linear group (mod. ra). If the degree is to be particularly noticed, we speak of the linear group of degree m k . It is however evident that all the substitutions of this group leave the element x , „ , . . . „ unchanged. For the congruences a x z x + bfy + . . . 4- 0,3*= 3, , a**, + b,z 2 -f & 2 z 2 + . . . + c 2 z k =z 2 ,. . . (mod. ra) have for every possible system of coefficients the solution Z! = 0, z 2 =0, . . . z A . = (mod. nt). We shall have occasion to employ the linear group in connection with the algebraic solution of equations. Theorem IV. The group of the geometric substitutions {mod. ra), or the linear group of degree m k is of the order given in 4). Its substitutions all leave the element a? >oj...o unchanged. It is commutative with the group of the arithmetic substitutions. ♦Galois: Liouville Journal (1) XI, 1846, p. 410. PART II. - APPLICATION OF THE THEORY OF SUBSTITUTIONS TO THE ALGEBRAIC EQUATIONS. CHAPTER EX. THE EQUATIONS OF THE SECOND, THIRD AND FOURTH DEGREES. GROUP OF AN EQUATION. RESOLVENTS. § 148. The problem of the algebraic solution of the equation of the second degree 1 ) x 1 — c r r + G 2 = can be stated in the following terms: From the elementary sym- metric functions c x and c, of the roots x t and x 2 of 1) it is required to determine the two- valued function a;, by the extraction of roots.* Now it is already known to us (Chapter I, § 13) that there is always a two-valued function, the square of which, viz., the discriminant, is single-valued. In the present case we have J = (a?, — x 2 f = (x x + x 2 ) 2 — ±x x x. 2 = c, 2 — 4r , . f^A — {x x — x 2 ) =\Zc } 2 — 4c 2 . Since there is only one family of two-valued functions, every such function can be rationally expressed in terms of hj A. For the linear two-valued functions we have a lXl 4- a 2 x 2 = — _— (a;, + x 2 ) -\ _- — fa— a*) and in particular, for a, = 1, a 2 — 0, and for a, = 0, a. 2 = 1 "2 — c i ~r — 2 — ' 2 ' •'i = S +2-VC, 2 - -4c 2 , Xa= -± — %*/c?--4& 2 . *C- G. J. Jacobi: Observatiunculae ad theorlam aequationum pertincntes. Werke, Vol. Ill; p. 2G9. Also J. L.Lagrange: Reflexions but la resolution algeorlque des equa- tions. Oeuvres. t. III.p. 205. ELEMENTARY CASES — GROUP OF AN EQUATION RESOLVENTS. 169 § 149. In the case of the equation of the third degree the solution requires not merely the determination of the three- valued function cc, , but that of the three three-valued functions x u ■>:, , .»■,. With these the 3! -valued function ? = «1#1 + «2 + «:!■'•; is also known, and conversely x x ,x 2 ,x z can be rationally expressed in terms of r. We have therefore to find a means of passing from the one- valued functions c u c 2 ,c z to a six-valued function by the extraction of roots. In the first place the square root of the discriminant J = (x 1 — oc 2 )' i (x 1 — x z ) 2 (x 2 — #3)" = — 27c 3 2 + 18030^! — 4c 3 c, :f — 4co 3 + <" ■' furnishes the two-valued function ± \%i X 2 ) (Xi X 3 ) (.*'o x 3 ), in terms of which all the two-valued functions are rationally expres- sible. The question then becomes whether there is any multiple- valued function of which a power is two -valued. This question has already been answered in Chapter III, § 59. The six-valued func- tion f— l+V^^l 9>i = «i + "'Xo + ojx 3 10 — ^ — —J cm being raised to the third power, gives we have 92 s = (a?i + «*£c a + o/ 2 x 3 ) = I (5, — 3 V — 3J) . Accordingly a?, + w 2 a- 2 + wa- 3 = v ^(^ + 3 V — 3 A , «i + "«a -f- w 2 a- 3 = V' i (5, — 3 V — 3i. Combining with these the equation *^i l~ Xo \~ x> 3 — C| , 170 THEORY OF SUBSTITUTIONS. and observing that 1 + w + or = 0, we obtain the following results •*'. = k I c x + % \(S t + 3 \/~SJ ) + i/ 1(^,-3 V^Sl)]' .r, = I [ Cl + w V | (flf, + 3 V — 3 J) + "-' v / 1 (5,-3 V — 3 A)] , x a = : \ [ c, + r» a V' l(S I + 3V = Sj+ w ^1(5, — 3\7==3J)1 • The solution of the equation of the third degree is then complete. § 1 50. In the case of the equation of the fourth degree it is again only the one-valued functions c u c 2j c 9 , c 4 that are known. From these we have to obtain the four four- valued functions x x ,x 2 ,x 3 ,x i , and with them the 24- valued function by the repeated extraction of roots. In the first place the square root of a rational integral function of c l ,c 2 ,c 3 ,c i furnishes the two-valued function ^/j. Again, we have met in § 59 with a six-valued function

2 by extraction of a square root, belongs to the group H=[l,(x 1 x 2 )(x a x t )']', /> = 12, r = 2. :,. . . . .r,) belonging to G. Every function belong- ing to G is rationally known and conversely every rationally known function belongs to G. Theorem I. Every special or affect equation is character- ized by a group G, or l>y a single relation between the roots '/'(a\,x,, . . . .»■„) = 0. The group (t is called the Galois group of the equation. Ever)/ equation is accordingly completely defined by the system — C\ — ( '\ j —X\Xn — o_. , . . . ; " {X l , X 2 , . . . JC„) := ( '. * ('f. Kronecker; Grundziige einer aritlnnetischen Theorie der algebraischen Gros- ser), S§ 10, 11. 174 THEORY OF SUBSTITUTIONS. For example, given a quadratic equation x 2 — dx-\-c 2 = 0. the corresponding Galois resolvent is 5" _. 2(«, + ajcf + (a, — « 2 ) 2 c 2 + 4 W, 8 = 0. In general the latter equation is irreducible, and the quadratic equa- tion has no affect. But, if we take 2c l = m-\-n, c. 2 — mn, the equation in I becomes (r ~a x m — « 2 n)(^ — a i n — a 2 m ) = =(•? — a 1 x l — a 2 x. 2 )(:- «,#<> — « 2 iC,) and the given quadratic equation has an affect. Again, if c, — c 2 = 0, we have ( : — ttjCj — o. 2 c.,y — = (I — a v T\ — a 2 a? a ) ( - — «iiCa — a -" T l )• But if c, — 2c, 2 = 0, we obtain ,- 2 — 2(«, + «,) c,l + 2 (a, 2 + «, 2 ) Cl 2 = 0, and this equation has no affect, so long as we deal only with real quantities. If however we regard i = \/ - - 1 as known, the equa- tion h# s an affect, for the Galois resolvent then becomes (f — (a, -+■ aj) c, + («, — a 2 ) C,t) (I — («! + « 2 ) C, — (a, — a 2 ) c,i) = 0. § 154. It is clear that every unsymmetric equation

*(*)] = o have then one root, a?, , in common, and since f(x) is irreducible, all the other roots x., , .r :! , . . . x„ of the first equation are also roots of the second. Consequently f(x) = is satisfied by {x x ), THEORY OF SUBSTITUTIONS. This group !-' is the group of the given equation. For the rela- tions which characterize the given equations are equivalent to the single relation and if this function <1> is to remain unaltered, then when .r, is replaced by x y , every .r must be replaced by

exactly as under the application of the substitutions of Q. £ 156. Without entering further into the theory of the group of an equation we can still give here two of the most important theorems. Theorem II. The group of an irreducible equation is transitive. Conversely, if the group of an equation is transitive, the equation is irreducible. Thus if the group G of the equation ./'(.'•) (x a?,) (a .<•,) ...<.»• --.<•„) = is intransitive, suppose that it connects only the elements a*, . x, »„ with one another. Then the function

(x) of the above form will be rationally known, and G can contain no substitution which replaces any element x lf x 2 , ■ ■ x a by x a ,, for otherwise the rationally known function c would not remain unchanged for all the substitutions of G. Consequently G is intransitive. Theorem III. If all the roots of an irreducible equation are rational functions of any one among them, the order of the group of lh" equation is u. Conversely, if the group of an equa- tion is transitive, and if its order and degree, are eijual. then all the roots of the equation are rational functions of any one among Hum. The lirst part of the theorem follows at once from $ 155. We proceed to prove the second part. ELEMENTARY CASES — GROUP OF AN EQUATION RESOLVENTS. 177 From the transitivity of the group follows the irreducibility of the equation. If we specialize the given equation by adjoining to it the family belonging to .r, , the group will be correspondingly reduced. It will in fact then contain only substitutions which leave x, un- changed. But as the group is of the type 11 (§ 129), it contains only one substitution, identity, which leave x l unchanged. Accordingly after the adjunction of as, , all functions belonging to the group 1 or to any larger group are rationally known. In particular .<•,, .r,, . . . x„ are rationally known, i. e., they are rational functions of x x . From this follows again the theorem which has already been proved in § 155: Theorem VI. If all the roots of an irreducible equation are rational functions of any one among them, they are rational functions of every one among them. § 157. From Theorem III, the group of the Galois resolvent equation of a general equation is of order n! To obtain it, we apply to the values ~\i *2J • • • '*! all the substitutions of .r, , .v,, . . . x„ and regard the resulting rear- rangements as substitutions of the ?'s . Since every substitution of the .r's affects all of the r's, the group of the £'s belongs to the groups £. The group of the .r's and that of the r's are simply isomorphic (§ 72). For an example we may take again the case of the equation of the third degree. The groups G of f(x) = and /' of F (r) = are then G=[l, f.<■■•<•,). (./',..-,.•.) | r=\_l, (fj? 3 ) (?2**) (^e)j <- r i"h) (-J-j) (~:i~+K Kl-"j) ("3-5/ (~4-fi)i Ul"|--,» I - _'~3~tt)' ( r i r .-.~4> ("-.'-h^sjj- If however, the given equation is an affect equation with a group G of order r, then of the n ! substitutions among the r's only those are to be retained which connect any =■, with those r's which together with r ; belong to one of thp rational irreducible factors F,(c) of F{=). 12 178 THEOKY OF SUBSTITUTIONS. i> 1 r>S. We apply the name resolvent generally to every //-val- ued function cm .»•,,.<■, «■„) of the roots of a given equation f{x) = 0. The equation of the ,""' degree which is satisfied by (a v + •'' ■' ',' + w-(a-,a- 4 + x 2 x 3 ), 3) The 12- valued function tb degree if ■ A l ^ ■>' ■ • ■ fp are the /'-values of (i)=P, and consequently one of the two integral factors, for example c(l), must be equal to ± 1. Moreover, since

a ) must vanish. Consequently p(a, 1 ) 9 (u>*) 1 *- 1 )=0 where a>, may be any root of 1), since the series 2) is identical with the series i,iOi,to*, ■ • ■ "V '• The equation 9 {x) ) ci.ncM./-) . . . en-" l )=F(x)(x p - 1 +x* -+...+.r+l), where F(x) is an integral function with integral coefficients. From 3 1 we have for .<• = 1 IX1)]*-' =P /''»*~ 1 . Since I) is irreducible, the corresponding group is transitive. There is therefore a substitution present which replaces ut 9 by or 1 '. Then every «j ya is replaced by ( ( o 9 ) 9a = w 9 ' l+ \ and the substitution is therefore s = (w 9 oj 9 "- w 9 ' . . . uj 9P ~' 1 ). The p — 1 powers of s form the group of 1 ). For they all occur in this group, and from § 156 the group contains only p -1 substitu- tions in all. We form now the cyclical resolvent (a> + au> 9 +a 2 a> 9 *-\- . . . -p-a*- 2 ^* -2 )*- 1 , in which a denotes a primitive root of the equation For brevity we write with Jacobi 0,8° _|_ am 9 ' + a 2 uj 9 " + . . . + a»~ V^~' = («, ai). From § 129 the resolvent («, uj) p ~ i is unchanged by s and its pow- ers, that is, by the group of the equation. It can therefore be expressed as a rational function of a and the coefficients of 1). If we denote a (p — l) th root of this rationally known quantity by ~, we have 5) (a, «)=T,. The quantity r, is a (p — 1)- valued function of the roots of 1 ). It is changed by every substitution of the group, for the substitution s converts it into (a, id 9 ) = a~ l (a, at) = a~ 1 t 1 . 182 THEORY OF SUBSTITUTIONS. It follows from the general theory of groups that every function of the roots can be rationally expressed in terms of r, . We will however give a special investigation for this particular case. The group of the cyclotomic equation leaves the value of 6) (o A , to) (a, o»)*- 1_A unchanged; for the effect of the substitution *.- is to convert this function into = (a A , w) (a, to)*- 1 -* i. e., into itself. If now we denote the rationally known value 6) by T A , where in particular ~ l p ^ 1 = T, , we obtain for / = 1, 2, . . . p — 2, the following series of equations: («» = *,, («?■,•) = ^V, («>) = |^.. :(«*->) =^t*->. Combining with these the obvious relation among the roots and the coefficients of 1) (1,") = -1, we obtain by proper linear combinations It is evident that a change in the choice of the particular root a p— i _ or of the particular value of r, = \/ T, only interchanges the val- in' w among themselves. Theorem II. The solution of the cyclotomic equation for the prime number p requires only the determination of a primitive root of the equation z p ' 1 — 1 = 0, and the extraction of the (p — l)' h root of an expression which is then rationally knoum. The cyclotomic equation therefore reduces to two binomial equa- tions of degree p — 1. THE CYCLOTOMIC EQUATIONS. 183 § 163. The second of these operations can be still further sim- plified. The quantity 7 1 , is in general complex and of the form Since now (a, id) 1 '- 1 and (« _1 , w ')'' ' are conjugate values, it fol- lows that (a, u>) p - l (a~\ w -1 )-'' -1 = p (cos ft -\-isin&)p(co8& — i sin ft) = //"'. Again it can be shown, exactly as in the preceding Section, that (a,iu) (a- 1 ,.""" 1 ) belongs to the group of the cyclotomic equation and is conse- quently a rational function of a and of the coefficients of 1). If we denote its value by U we have V p = v u. Accordingly for any integral value of A; r 0-f-2*7T , . . ft + 2k-\ (a. u) ) = v U I cos - —z — \- 1 sin — I v p — 1 p — 1 J. Since U and ft are both known, we have then Theorem III. The solution of the cyclotomic equation requires the determination of a primitive root of the equation z p ~ 1 — 1 =0, the division into p — 1 equal parts of an angle which is then known, and the extraction of the square root of a knoivn quantity. The latter quantity, U, is readily calculated. We have U= (u> + a to* -+- o 2 w""- + . . . + a*- a ai« rl>-a ). (< U - 1 -f-a- 1 w-« , + a- 2 ttf-» + . . . +a-*+*a>-<> p - v ). To reduce this product we begin by multiplying each pair of cor- responding terms of the two parentheses together. The result is 1+1 + 1+.- .+1 =p-l. Again, if we multiply every term of the first parenthesis by the k th term to the right of the corresponding term in the second par- enthesis, we obtain K) a -*(a»-'*+ , + e»--»* +1 +' + a»-'* +r +'* + . . .). Now w^ ffk + 1 is a i» tb root of unity <«, different from 1; for if 184 THEORY OF SUBSTITUTIONS. then -0*+1e=O, >-+ -*+2) =p _ 1 _(_ 1 ) = p Theorem IV. The quantity of Theorem Til, of which the square root is to be extracted, has the value p. §164. The resolvent 5) was (p — 1 (valued, and consequently the preceding method furnished at once the complete solution of the cyclotomic equation. By the aid of resolvents with smaller numbers of values, the solution of the equation can be divided into its simplest component operations. Suppose that p x is a prime factor of p — 1, and that p — 1 =p l q ] . We form then thr resolvent (a) -j- «,a»» + a, V' 2 -f . . . + a* -or' 1 ' '-)"., where a, is a primitive root of the equation S l>i_l = 0. The values . cr, = or' -f (0**1 + ' -f <''• - ')*| . the resolvent above can be written ( i + a iVa + . . . + «i*» ' ?„ - - 1)* 1 , or, again in Jacobi's notation, By the same method as before we can show that this resolvent is unchanged if "> is replaced by <»'■', that is, that it belongs to the group of 1) and is consequently a rational function of a, and of the THE CYCLOTOMIC EQUATIONS. 185 coefficients of 1). We denote its value by A', = >,''i, and have accordingly ("i , is replaced by <" . that is. they are unchanged by the subgroup s'\ n-''i. S :, "i, . . . 8*1*1. We have therefore Theorem V. The p x -valued resolvents

y determining a primitive root of z -1 = and extracting a p.,"' root of a quantity which is rational in this primitive root and in o is given, -"">) = 0, 7) I (X — W) (X — <0° P 1 +1 ) (x — o^>': i ') . . . (X — W«<* '"'' + ') = , are all rationally known. Accordingly after the process of Theo- rem V has been carried out, the equation 1) breaks up into p, fac- tors 7). Since the group belonging to each of these new equations is transitive in the corresponding roots, the factors 1) are again all irreducible, so long as only the coefficients of 1) and

»* p *) («—«.•*>*) . . . {x— m'to- 1 ***) = 8) are rationally known. The equations 7) are therefore now reduci- ble, and each of them resolves into p 2 factors 8), which are again irreducible within the domain defined by /,,. We can proceed in this way until we arrive at equations of the first degree. § 166. The particular case for which the prime factors of p — 1 are all equal to 2 is of especial interest. Theorem VIII. // 2'" + 1 is a prime number p, the cyclo- tomic equation belonging to p can be solved by means of a series of m quadratic equations. In this case the regular polygon of p = 2"' + 1 sides can be constructed by means of ruler and compass. In fact, for one root of the cyclotomic equation we have 2- , . . 2- _, 2- . . 2tt io = cos \-ism — , = cos ism — , P P P P . 2- u)-\- o> ' = 2 cos — , p 2- and consequently the angle : u can be constructed with ruler and compass. In order that 2'" + 1 may be a prime number, it is necessary that m = 2*. For if m = 2' t m 1 , where m, is odd, then 2"' + 1 = (2 2M )'" 1 + 1 would be divisible by 2 -j- 1. If # fi = U, 1, 2, 3, 4, the values of p are actually prime numbers p = 3, 5, 17, 257, 65537, and in these cases the corresponding regular polygons can there- fore be constructed. But for ,« = 5 we have 2^ + 1 = 4294967297 = 641 • 6700417, L88 THF.OKY OF SUBSTITUTIONS. so that it remains uncertain whether the form 2 a -f- 1 furnishes an infinite series of prime numbei § 167. We add the actual geometrical constructions for the caM>- p 5 and p = 17. For p = 5, we take for a primitive root g = 2, and obtain accord- ingly g° = l, g l = 2, ;/' = 4, g 3 = 3 (mod. 5). Consequently i 2 i ; V'n — - °' ~T~ '" • < P\~ Z "' T '" n + '• (o = ens _ -J- ■ / SMi = , ') then 2- ;- c'„ = a» -j- «/ = 2 ro.x . . Vi = or -f- tt> 8 = 2 COS -= . D consequently cr„ > 0, ^ < 0, and the \/5 in the expression above is to be taken positive. Furthermore =="'; z, — u>-, / :! = o/' r— ^,/ + l = 0; -i + V5+*Vio+2 \71 /., to =■ -i+ V5-;Vio+ 2Vt> Zl=w = . — , •Cy. Gauss: DIsquiBit. arlthm., § 362. The statement there made that Fermat sup- posed all the numbers •i- 1 '- + i to be prime, is corrected by Baltzer: Crelle 87, p. 172. At present the following exceptional cases are known: T | 1 divisible by 641 .Landry), ,12 t i divisible by 114689 (J. Pervouchine), i divisible by 167772161... 'J. Pervouchine; E. Lucas . i divisible by 274877906945 (P. Seelhofl), Cf. P. Beelhoff: Bchlomllch Zeitschrift. XXXI. pp. 172-4. THE CYCLOTOMIC EQUATIONS. 189 the sign of i being so taken that the imaginary part of to is positive and that of c/ negative. c • H ,-" E\ ' For the construction of the regular pentagon it is sufficient to know the re- . 2- solvent cr = 2 cos -g . D Suppose a circle of radi- us 1 to be described about O as a center. On the tan- gent at the extremity of the horizontal radius OA a distance AE = %AO=tt is laid off. Then oe= VT+i = -o- ■j If now we take E F ' = E O, we have V5 — 1 AF-EO— EA = — ^ = ?oi 2- AF= 2cos-=-. 5 Finally if if is bisected in G and GffJ drawn to Oi and 2- OC I to JEfJ, then HOC=COJ—-^-, since cosHOC = AG = co.s~"~ . if, C, and J are therefore three successsive vertices of a o regular pentagon. § 168. For p = 17 we take for a primitive root g = 6. Accord- ingly we have o , o x \ g 11 , flfVs 18 , g u , o l \ a + to* + «/ + w lu + w 15 + w 13 + w\ ?i —u, 6 -\- w u -f oi -f a» 14 + w 11 + 10 + u> J ; Vo + 9i = — 1- 190 THEORY OF SUBSTITUTIONS. To rind o + fi + fi + ?i + 9o = 4 (? + Pi) = — 4. Consequently ?'o+? , i = — 1) 9>o9i = — 4 r + v — -1 = 0, -1+VT7 -l — V17 fl 2 ' ri 2 where the sign of V 17 is undetermined until the particular root (a is specified. If we take 2* , . . 2* m— COSir=-\-lStnj=, we have for the determination of the sign cr, = (o> 3 + w M ) + (o» 5 + w 12 ) + (a* 8 + » u ) + (o> 7 + to 10 ) r 6* io,r 12* 14*1 = 2 [cos j= + cos ^- -f- cos ^ + ro.s - J ,[ r,- 7- 5* 3*1 = 21 COS -= COS -= — COS p=r — COS -r „- J ^ .:.9i<0 t and Vl7 above must be taken positive. We have further /o = a, + w * + w '« _|_ »» £ = a. 2 + w s + ai" -f a» 9 : Zj = a» 8 -f w 7 + a>" + w 10 , /, = w 1 " + o, u -f w D -}- at' ; /../, -=/.;+/. +/o + /,.= — 1, XiXz—Xa +X» +*2 +*i = — lj X*--nX — 1 = 0, f — 9\X — 1=0; *>_, / yo 2 + 4 _ft. I c'-, , and there- fore C'<„=^4- /ft v ,', - ^ Z "" v '° 2^a/T~~ /3 ' 01 ""T""V"2"" These results suffice for the construction of the regular polygon of 17 sides. Suppose a circle of radius 1 to be described about O as a center, and a tangent to be drawn at the extremity of the hori- zontal radius OA. On the tangent take a length AE = \OA — \: then \/17 OE= Vl-r-A = Further, if EF—EF' = EO,we have A ^- 4 ~2' 4 192 THEORY OF SlTBKTITrTIONS. OF: ^ *L + 1, OF' ^ £ + 1. Taking then we have FU — FO, F'H' = F'0, n AH=AF+FO f + TJ : f+ 1 =Xo, n AH ' = — A F' + F'O - ^ + J£ +1 =Xs< We bisect A if in )': then AY=fa. THE CTCLOTOMIO EQUATIONS. 193 We take now AS = 1. and describe a semicircle on H'S as a diam- eter; if this meets the continuation of OA in K, then AK 2 = AS- AH' =/,. Again if we take LK = AY and KL = LM = LN, and describe a circle of- radius LK about L, we obtain AN+AM = 2KL = 2AY= Xo =

, = 2, under the assumption p > 2. If g is a primitive root, then tr |1 = u) -f- to"' -f- '" -)-...-)- o) 9 , , = io 9 -f- <> ; -f- <>' -\- . . . -f- a) 9l '~. 2 , l) and in the third p — l case — =p- . ' 13 iy I THB0KT OF SUBSTITUTIONS. Consequent 1\ S) to, -\- m. 4- ?». = — - — , where ///,, m., m , represent the number of brackets of the several kinds. If />;, = 1 according as — - — is even or odd. Since ' ) that (mi -f- w) p -j- ( //t_, + ?i ) cTj = ( I In this equation all the powers of <» can be reduced to powers lower than the p th , and the equation can then be divided by <». The resulting equation is then of degree p — 2 at the highest, and still has the root to in common with the equation 1 ) of degree p 1. It is therefore an identity and 77i, = m.> = - — n Consequently we have for the values of ///,. m , and cr n , cr, p — 1 /'— 1 fP — 1 \ m,=.m.,= -j—, cp ^, = - — j— , (— ^— even) p t3 p — l p— 3 p + i rp— l «/,=TO 2 = — , ^0^1 = t-',, cr. 2 4 4 y-i _!-(-!) ' P [ , (^ odd) THE CYCLOTOMIC EQUATIONS. 195 P — {*) ...(.c — w ; ' J ' I = 0, Zi^CC — a> 9 ) (X — Oi 9 *) (x — co' j: ) . . . (x- co r ' P ^' ! ) — (). The roots and consequently the coefficients of these equations are un- changed if w is replaced by . In particular x\ = 0{x x ) must be a root of 3), so that f\o[e( Xl m=0. Consequently 0[#(#,)J is a root of 1) and therefore of 3). Then |0[0(#i)] \ is a root of 1), and so on. With the notation d [d {x)-] = 2 (x)] , [d> (a;)] = 2 [0 (* )] = 6* (x), . . . . , it appears that all the members of the infinite series <*> *(»i), 6 \^), ^(*i), ••• 6 \*i), ••• are roots of the equation 1). Since however the latter has only a finite number of roots it follows from a familiar process of reason- 198 THEORY OF SUBSTITUTIONS. ing that there must be in the series a function ^"'(cc,) which is equal to the initial value x t , while all the preceding functions .r i ,0(x 1 ),0\ Xl ),...0'"- 1 (x l ) are different from one another. The continuation of the series then reproduces these same values in the same order, so that only F»(x l ) = 6 2 ' n (x 1 ) = . . . take the initial value x A , and that for k < m only ir i-*^) = 0*" *(#,)= . . . are equal to t> k '(x^). If the system of in roots thus obtained does not include all the roots of the equation 1 ), suppose that .v., is any remaining root. Then x, also satisfies 3), and therefore 0(x 2 ) is a root of 1), and so on. Accordingly we have now a new system of :>■ different roots X ii 6{x i ),e\x 2 \...d^~\x 2 ). But since the equations 4) ff m (y)—y = W(z)—Z = Q have each one root y = x x , z = x., in common with the irreducible equation 1), they are satisfied by all the roots of the latter. The former equation of 4) is therefore satisfied by .r L ,, the latter by a;,, consequently tn is a multiple of fi and />. is a multiple of to, L e. m — v. Again all the roots of the second series are different from those of the first. For if ffi(x i ) = P(x i ) (a,b systems of m roots each, as in the following table : 5) THE ABELIAN EQUATIONS. 19W x u 0(0;,), Pfa), ...e m -\xj, x 2 , B(x i ), » J i u-.). . .." l (a? 3 ), x v , n(x v ). "' ~ l {x r ). The function 6 is such that for every root x a m (x a ) = x a , and the equation f (x) = is of degree m>. § 172. We can now determine the group of the equation 1). Since the equation is irreducible, its group is transitive (§ 156); it therefore contains at least one substitution which replaces Xj by any arbitrary x a . It follows theu that all the roots of the first line of 5) are replaced by those of the « th line. The group of 1) is there- fore non-primitive and has v systems of non-primitivity of m ele- ments each. The number of admissible permutations of the v systems is not as yet determinate; in the most general case there are \>\ of them. If any x a is replaced by A (a? a ), then every "*"(.*■„) is replaced by 0* +A (a? a ); there are therefore m possible substitutions within the single system. The order of the group of 1) is therefore a multiple of m v and a divisor of v\m v . Theorem II. The grotvp of the equation 1) is non-jprimitive. It contains v st/stems of non-primitivity, which correspond to the several lines of 5). The order of the group is r = r, m", where r, is a divisor of v\, § 173. In the following treatment we employ again the notation of Jacobi z 4- toz 1 + to-z 2 -f- . . . + to m ~ l z m _, = (<«, z), where m is a root of the equation w m — 1 = 0. Similarly we write ». + »%«) + «'«■(»-)+ • • • +<» , "- 1 (>" , -\x a ) = (w, d(x a )). We form then the following resolvents: 9l = (i , o( Xl ) ), n = (i , o(&) ),■■■ ?v = (i , *M ) • cr, is symmetric in the elements of the first system and is changed in value only when the first system is replaced by another; it is therefore a v- valued function, its values being c x , cr,, . . . cr„. 200 THEORY OF SUBSTITUTIONS. Every symmetric function of the c's is a rational function of the coefficients of 1 ) . The quantities that is, the coefficients of the equation of the v th degree o) r— s 1 9> , - 1 +Stf v - s — . . . ±s y =o, of which c-,. cr„, . . . cr,. are the roots, are therefore rationally known. Theorem III. The resolvent ( 1, i>(x,) ) = sb, + »(x x ) + &(x x ) + . . . + »'" ~ '(.<•, ) is a root of an equation of degree >, the coefficients of which are rationally known in terms of the coefficients of the equation f(x) = 0. i> 174. The equation 6) has no affect (§158), unless further relations are explicitly assigned among the roots x n x 2 , • • • x n . If however any root c a of (5) has in any way been determined, the values of the corresponding .r a , n (.r a ), . . . can be obtained algebrai- cally by exactly the same method as that employed in the preceding Chapter. Thus, the equation #) of which the roots are x, 0(x), 0\x), . . . m ~\x) is irreducible. Its group consists of the powers of the sub- stitution (\0() 2 . . . (J"'- 1 ). And if we write where w is now assumed to be a primitive m th root of unity, we have [O(x)+wO%x)+...+ m - 1 m (x)~\ = (a>,o(x)-\»; that is T, is unchanged by the substitution (\O0 2 . . . fl m - '). Conse- quently T, is a rational function of the coefficients of #) and of the primitive m th root of unity to. The m tb root of this known quantity 1\ we denote by r, . Again if we write (oj\0(xj)(aj,0(x)r-*=T x , it can be shown by the same method that we have already frequently THE ABELIAN EQUATIONS. 201 employed that r l\ is also rational in o> and the coefficients of &). Taking successively /. = 0, 1.2.... m— r l, and combining the result- ing equations, we have then rji m0(x) = cr + -r, + - 2 t V+ " 8 S m 8 +•••+'■ w "Mi The function r x being m-valued, a; also admits only m values, and these coincide with x , 0(x), 2 (x) , . . . . If any other m th root of T x is substituted for t, , the m values of x are permuted cyclically. Theorem IV. If the m roots of an equation of degree m are where 0(x) is a rational function for which m (x 1 ) = x } , then the solution of the equation requires only the determination of a primi- tive root of z"' — 1 = and the extraction of the m?' root of a known quantity. Theorem V. If one root of an equation of prime degree is a rational function of another root, the equation can be solved algebraically. For in this case we have m>=p and m > 1 ; consequently m=p and v = 1. § 175. If all the coefficients of f(x) are real, the process of the preceding Section admits of further reduction. The quantity T x — (a* , 0(x))'" = p(cos # + isinQ) can be rationally expressed in terms of u> and the coefficients of /. The latter being real, the occurrence of * = V — 1 in. ^i is d u © .entirely to the presence of o>. Consequently. T,' = (io-\ 0(x) ) m - p(cos & — i sin &), T& = f, is* 202 THEORY OF SUBSTITUTIONS. where U is again rationally known, since it is unchanged by the group of c . We have then. r l = V U\ cos \-iHin — ! I. V. m ni J Theorem VI. If all the coefficients of f(x) are real, the second operation of Theorem IV can be replaced by the extraction of a square root of a known quantity and the division of a known angle into m equal parts. § 176. If the m of the Theorem IV is a compound number, the solution can be divided into a series of steps by the aid of special resolvents. If m = to, to/, where to, is any arbitrary factor of to, we take 0, = xx + #'"' (a;,) + (Xj) , tp 3 = 0{x,) + u'"i + ' (x x ) + 2, "> + 1 (x,) + . . . + tf('"'> - ») '"> + ' (a;,) , V '.„ (] = 0-1 ~ » fa) + *■"* - : (as,) + 3 '" 1 - 'fa) + . . . + "(.£,) = 1] and if in = m 1 m 2 m 3 . . . , the sohdion of the equation requires only the determination of a primitive root of each of the equations g m,-i_0 } £».,_! _o ? 2 m 8_l = 0, M . and the successive extraction of the m," 1 , m 2 th , m 3 th , . . . roots of expressions, each of ivhich is rationally known in terms of the pre- ceding results. § 177. The solution can also be accomplished by a still different method. Suppose that m = m x nu . . . m u = m,tfi x = m. 2 n 2 = . . . = m 01 ti M . Then we can form the following equations : g,(x)=Q, with the roots x u o n, Ux t ). e^(x 1 ), . . . d^-^fa), and with coefficients which are rational functions of a resol- vent /, = ,r, + "'"^ (Xi) -\~ ■ . . /i is a root of an equation » 1>i ('/) = of degree m, . C g,(x) = 0, with the roots x l ,0 m *(xj,P m »(xJ t . . . rt^" *>%(*,), and with coefficients which are rational functions of a resol- vent Xi — x i + V'"Hx } ) + . . . /.. is a root of an equation h 2 (b/) = of degree m, . A,) A 2 ) Aj g m (x) = 0, with the roots ; »,, 0"' a '(x i ),o^(x i ), . . . • • • P* are the different prime factors of m , then we are to take m i ~P a, y m 2 = P% • • • m "{x) = x. The values of x which are unchanged by the operation satisfy the equation ax-}- j3 X ~ yx + d' yx 2 -f- (d — a) x — ,5 = 0. For these fixed values we have therefore, according as ad — t 3y = ± 1, and consequently yx yx ' — a Ya + d Ifa + d'V , V A) We assume in the first instance that x' and x" are distinct, that is that iV-]-l. We have then d( x )—x' _ N 9{x)—x" 0"'(x)—x' i)'"{x)—x" = N x — x' 2 (x)- -x' — N 2X X II 5 X — X x—x"' m x — x' e\x)- -x" x — x" The necessary and sufficient condition that m {x) = x is there- fore that '200 THEORY OF SUBSTITUTIONS. This condition can be satisfied by complex or by real values of the quantity in the bracket. In the former case the upper alge- braic sign must be taken, and further so that we may write a + d —^— = cos '(■>■) = x, which agrees with the condition 9), since in — 2. B) It remains to consider the case x' = x". We have then ■a + d i T The lower sign must be taken, and accordingly a + d = ±2, ad -fr=+k It follows that m±i-* o\x) - (2a Tl).r + 2/5 2r#+(2d + 1)' e\x) (3a =F2)x : + (33 ; 3/S 1 2)' /'(,■, | ma =F (m — 1> + mft If now w'"(x) = x, we have THE ABELIAN EQUATIONS. 207 r a* + (d—a)x—p = 0, that is, we must have already had 9{x) = x. And, again, it is clear that, as m increases. (T(x) approaches the limiting value We have shown therefore that k~ a 4- d = 2 cos — , "•'» — ,J/' = 4- 1 , ra where A; is prime to m, are the sufficient and necessary conditions that (T(x) shall be the first of the functions K (x) which takes the initial value x. For m = 2 , the second condition is not required. § 179. For the second example we take for 0(x) any integral rational function of x with constant coefficients. For every integral m the difference m (x) — x is divisible by 0(x) — x. For if 0(x) — x= (x — z 1 )(x — z 2 ) . . . (x — z v ), then for every z a 0(z a ) =z a , and consequently 2 (z a ) = 3 (z a ) = . . . = k (z a ) = z a . Moreover 6 k+i( x ) — e l (x) = [0*(a;) — z,] [G\x)—z.^ . . . \?(x)— z v ~\, and consequently k + \x) — 6 k (x) _ P(x)—z l k {x)—z 2 k (x v ) — z v _, . — — . ■ ( ^ ^ — ItjAQC) ? where P is a rational integral function of x and of the coefficients of 0, since it is symmetric in the roots z lt %, . . . z v . If now we take k successively equal to 0, 1, 2, . . . m — 1, and add the resulting equations, we have as asserted 0"'(x)—x = [0(x)—x] Q(x), where Q is a rational integral function of x . From this equation it follows that for every root of Q(x) = we have m (x) = x, and conversely that every root of m (x)—x = O, 208 THEORY OF SUBSTITUTIONS. which is not contained among z,, z.,, . . . z v also makes Q(x) vanish. Every root r of Q(x) = therefore gives and consequently also 0'" +!(£) = 0(£), flr[0(£) ] = (?| 5 ). so that #(c), and likewise 2 (£), # 3 (f), ... are all roots of #(x) = 0. Again since = is different from the z's, 0(5) -f- 1, and fl(|) ==z tt . Theorem IX. If 0(x) is a rational integral function of x of degree >■> , then the roots of the equation of degree (v — l)ra caw be arranged as in Theorem I, 5). If m is a prime number then each of the v — 1 rows of 5) contains m roots *,m *(*>,•• -i*- 1 ® p-w^o § 180. Conversely if the equation / (x) = has the roots x ,x 1 = 0(x ), x 2 = 6 2 (x ), ...x m _ l =6 m ~ l (x,) ; [(> m (x ) = .r„] . every one of these roots will also satisfy the equation m (x)— x = 0, but no one of them will satisfy b(x) — x = 0; consequently / (x) is a divisor of the quotient H m (x) — x o(x) — x The restriction that e(x) shall be an integral function is unessen- tial. For if (x) is fractional 6{X) — t— ; , where a, and g 2 are integral functions, then in g(x x _ gi(a?o)[ga(agi)flf2(a?a) • • • ga(a?«.-i)] 2 (#o) 92(^1) gi(pt: a )...gi(x m _ l ) the denominator, being a symmetric function of the roots of /(#) = 0, is a rational function of the coefficients of f(x); and the THE ABELIAN EQUATIONS. 209 second factor of the numerator, being symmetric ina; M x 2 , ... x m _ , is a rational integral function of x . Consequently 0(x ) is a rational integral function of x , which can be reduced to the (m — l) th degree by the aid of f(x ) = 0. We have therefore Theorem in X. Every polynomial of the equations treated in § 174 is a factor of an expression 9 l "'(x) — x x (x)—x ' where 9 x {x) is an integral function of the (m — l) th degree. For example, if we take X — x~ + bx + c , we may reduce this by the linear transformation y = x + « to the form (\ = x 2 + a. Then 9*1 x) —x = ( 1 (x)—x) |> 6 + x> + (3a + \)x k + (2a + l)x' + (3a 2 + 3a + l)x 2 + (a 2 + 2a + l)x + (a 3 + 2a 2 + a + 1) ]. The discriminant of the second factor on the right is J = — (4a + 7)(16a 2 + 4a + 7) 2 . If now we take 4a + / = — Ar, a = -r — , the second factor breaks up into two, and this is the only way in which such a reduction can be effected. We have then 0,(;r)— x [8.r 3 + 4(1 + k)x*— 2(9 — 2k + k 2 )x— (1 + 7fc— fc 2 + fc 3 )] [8ar j + 4(1 — k)x>— 2(9 + 2fc + k 2 )x—{l — lk—k 2 —k*y\ or, for fc = 2A + 1, a = — /- — / — 2, [y _ A ^ _ ^. + 2/ + 3)a + (^ + 2/-' + 3;. + 1 )] . In this way we obtain the general criterion for distinguishing those equations of the third degree the roots of which can be expressed by x, 0{x), 2 {x). In the first place 9 must ^be reducible to the form = x 2 — (A 2 + /1 + 2) 14 210 THEORY OF SUBSTITUTIONS. that is. b*— Ac must be of the form 4 (A 2 + /. + 2) = (2 A + 1)- 4- 7 ; then to every there correspond two equations of the required type x 4- (). 4- l)a* - (/.- 4- 2)x — (/* + /- 4- 2/ 4- 1) = .,. _ xrf _ (;a _|_ 2/ + Z) X + (A 8 4- 2A 2 + 3/ 4- 1) = 0. It appears at once, however, that <> is unchanged if / is replaced by —{X 4- 1), and that the first equation is converted into the second by this same substitution. It is sufficient therefore to retain onlv one of the two equations. § 181. We introduce now the following Definition. If all the roots of an equation are rational func- tions of a single one among them, then, if these rational relations are such that in every case the equation is called an u Abelian Equation." * We have already seen (§ 173) that, if the roots of an equation are defined by 5), the resolvents ft = (1, *,(*,) ), ft = (1, *iM >. ■ • • ft = (1, *i(«r) ) satisfy an equation 6) of degree v the coefficients of which are rationally known. We noted further that this equation is solvable only under special conditions. These conditions are realized in the present case. We proceed to prove Theorem XI. Abelian equations arc solvable algebraic- ally. ** In the first place we observe that since r l 0*a), and assume that jr.. = 2 (a;,), we have c 2 = e 2 { Xl ) + e t e 2 (xi) + ^%{-^) + • • • + K %{x, > = ^,) + ¥i(a'i) + ¥ I 2 (-' , 1 )+ • • • +W -\*i) = Sj>„ g.fo), ^O,), . . . ff," -'(j,)] = *(?,). * 0. Jordan : TraitC- etc. § 402. *»Abel: Oeuvres completes, I, No. XI; p. 114-140. THE ABELIAN EQUATIONS. 211 For from x a (x) = 0J x (x) follows also e;%{x) = O^O^x)] ■ O l e t [0 l {x)'] = OJcix);. . . The equation c, = R(

. A (X\) = 3 2 (#i )> it follows that st — ts. All the substitutions of the group are therefore commutative. If conversely the group G of an equation f(x) = consists of commutative substitutions, we consider first the case where G is intransitive and f(x) is accordingly reducible (§ 156). Suppose that A*) = /i(«0/.(*).-. where fi(x),f 2 (x), . . . are rationally known irreducible functions. If we consider the roots of /,(•»') =0 alone, every rationally known "J 12 THEORY OF SUBSTITUTIONS. function of these is unchanged by the group and conversely. Accordingly we obtain the group G, belonging to/,(;c)=0 by simply dropping from all the substitutions *, . s 2 , s 8 , ... of G those elements which are not roots of f(x) = 0, and retaining among the resulting substitutions , THE ABELIAN EQUATIONS. 213 in which every root occurs once and only once. The numbers »!, n 2 , . . . n k are such that every one of them is equal to or is con- tained in the preceding one, and that they are the smallest numbers for which 1 »(x 1 ) = x 1 , l ) = Xl ,...0> u *(x l ) = x l , respectively. There is only one substitution of the group of the equation which converts .r, into a {x^). Denoting this by s a , we can arrange the substitutions of the group also in a system s,**,**,*" . . . S*** (hi = 0, 1, 2. . . . n t — 1). Hiii,n 3 . . . n k = n, where again every substitution occurs once and only once, and cor- responding to the properties of the #'s, s 1 "i = l, */'- = l s fc B * = l. The numbers »,, n.,, . . . n h are the same as those for the fl's. To form a resolvent we take now *t(«a) = 2 °'' :l>> ■ ■ ■ < rH - Cl) (hi = °' *' • • • Ui _ 1} *S i *8 1 • • • h h and construct the cyclical function xfa) = [W^) + "x e M*i) + < e ?^*x) + • • • +■'> + • . • + <■ V« V, , ,(«0JS in which w 2 is a primitive h., 11 ' root of unity, the function /, _,(.<•,) is unchanged by the group G.,, and is therefore a rational function of ft ■ For the substitutions of the group leave 0, , , unchanged and the powers of s._. convert ft 1 2 into 0,ft 2 "A' 1 , - . . .6^-tyi respectively. Applying Theorem IV again, we obtain ft >a from ft by the solution of a second simplest Abelian equation of degree a .. In general, if we write Vl,8,... V— ^ £j >■ . I ... /, ^ij, #1,2, ... K = [^ I 2,...»+'»M lJ| ...,+'»AVi 1 2 1 ...,T...+<' '"/"•V',,,,...*']'"', the value of ft f 2,...v is determined from that of the similar func- tion ib = ^S ^ *v e h iA r ) Y\,2,...v-\ — Xj v ••• k k \->-\)- I'v . . . I'k by the aid of a simplest Abelian equation, as defined by Theo» rem IV. By a continued repetition of this process we obtain finally * Theorem XIII. If the n roots of an Abelian equation are defined by the system i*,) (/*,- =: 0, 1, 2, . . . n { — 1) the solution of the equation can be effected by solving successively k "simplest" Abelian equations of degrees n l ,n 2 ,n z , . . . n k . *L. Kronecker: Berl. Ber., Nachtrag z. Dezembcrheft, 1877; pp. 846-851. THE ABELIAN EQUATIONS. 215 § 184. The solution of irreducible Abelian equations can also be accomplished by another method, to which we now turn our attention. Theorem XIV. The solution of an irreducible Abelian equation of degree n = p l "ip., a ^ . . . , where Pi,p>, . . . are the differ- ent prime factors of n, can be reduced to that of k irreducible Abe- lian equations of degrees jVS iV% • • • The proof * is based on the consideration of the properties of the group of the equation. For simplicity we take n = Pi a ip 2 a --. Since the order of the group is r = n, the order of every one of the substitutions is a factor of n, and is therefore of the form Pi" l lh"-- Every substitution of the group can accordingly be con- structed by a combination of its (p 2 a -) th power, (which is of order pfi) and its (p^) th power (which is of order p 2 b "). Consequently we can obtain every substitution of the group G by combining all the t' f f f the orders of which are a power of p v , with all the /" /" t" t" the orders of which are a multiple of p.,. Since the Vq are all com- mutative, the substitutions of G are, then, all of the form The order of the product in the first parenthesis is a power of p x , and therefore a factor of p 1 a \. For we have Two substitutions . {t' a t' p ... )(W«,..), (t'j' b ...)(t" d t" e ...) are different unless the corresponding parentheses are equal each to each. For if the two substitutions are equal, we have (t'J'p ...) (t'J' b ...)-'= (t"*t"< ■ ■ .)-\t" d t" e . . .), and since the order of the left hand member is a divisor of p x \ and that of the right hand member a divisor of p 2 a -, each of these divi- sors is 1. •O. Jordan; Trait6 etc. § 405-407. 21G THEORY OF SUBSTITUTIONS. The number of substitutions ,s is equal to n = p, a i p a "«. And since the substitutions if form a group and every substitution of this group is of order p,"*!, the order of the group itself must be />,'">, (§ 43). Similarly the order of the group formed by the t' n & is equal to p"'-. It follows then from n = pfp.f* = p"' x p 2 ' n - that Wj = «, , m^ = a 2 . Suppose now that

belonging to the group of the t' 's is a root of an Abelian equation of degree p 2 a ^. If now

x - l ^t l ^' i t/^ l = l, so that t x U is also of order not exceeding p K '. If the group H is of order p", any function

)■ It follows then that s., = s.rp- b ) = s 1 m ( a - a \ which is contrary to hypothesis. Accordingly /S = b and a = a. If a > 2, suppose that s 3 is a substitution of G not contained among the p 1 substitution s{'s.!'. Since s,s 3 = s 3 s x and s 2 s 3 = s 3 s 2 , the group H z = -J.s,, s 33 s 3 (• contains at the most p 3 substitutions. And it contains exactly this number, for if s, "s. b s{ = s^s/s^, then s 3 y ~ e =s l "~ a s 2 b ~P, and so on, as before. Proceeding in this way, we perceive that all the substitutions of G can be written in the form s^sA . . . s a Xa , (X t = 0, 1, . . . p — 1) • 218 THEORY OF SUBSTITUTIONS. where every substitution occurs once and only once (cf. § 183). If now we take for the resolvents and the corresponding groups Va ' ''"l ? •'".' ? • • • •'">! ) i '•■ a - \ ,s 'l 5 , ',i) '1 "a- 1 — '1 -S 'l > ®2j • • ■ ®a 2 J ,s 'a i • P] ' 'm '''j '"..') -" 1 — '(' S -' S ^1 ' ' ' ®»J I then every resolvent depends on an Abelian equation of degree p. The roots of the given equation of degree p a are rational functions of Pi> Pi'> ■ ■ -fai for the function 4> = ft P] + ft Pi + • • • + 0a Pa belongs to the group l(c/. § 177). £ 187. The p a roots of such an equation may be denoted by . **,,**,...*„ (** = 0,l,2»...p— 1) Suppose that #$,,&,.„ £ a is the root by which s/i.s/s . . . s a ^° replaces x z , . . Then the substitution Sj^s 2 ^ 2 . . . sj a ■ s^s.^- . . . sj a = s 1 ^ + £>s 2 & + & . . . sj"-+ to- by virtue of the left hand form will replace .r., iC ,,... „ a by the root by which sj'sfi . . . sj a replaces #&,&,... £ tt . -^ u ^ ^ rom ^ e right hand form this root is ^.+|, ,^ + f, ... ^ a +f a - Consequently every substitution sfisj* . . . sj a replaces any element #£,,&,...$„ by »& + *!, & + &,.-. Sa + £a> that is the substitutions of the group are defined analytically by the formula •V'.s/'-' . . . s a *« = | 2j , z 2 . . '. z a z x + k x , z 2 + fc 2 , . . . z a + fc a | (mod. p). 77te group of an Abelian equation of degree p a , the substitutions of which are all of order p, consists of the arithmetic substitutions of degree j> a (mod. p). § 188. Finally we effect the transition from the investigations of the present Chapter to the more special questions of the preced- ing one. 2-' Let n be any arbitrary integer and let the quotient — be denoted by a. Then, as is well known, the n quantities THE ABELIA.N EQUATIONS. 219 cos a, cos 2a, cos da, ... cos na satisfy an equation, the coefficients of which are rational numbers, C) .r"~ \ noc" J + ,',, ^~^ a;"-*— ...=0. If now we write x= cos a, then for every integer m cos m a = v(cos a) , where d is a rational integral function. Similarly if the value cosi^a is denoted by dfaosa), we obtain, by replacing a by m^a in the last equation, the result cos(mniia) = tticosm^a) = B6 1 {cos a) Again if in the equatioL J (cosa) = cos(m 1 a) the argument a is replaced by ma, the result is cos^n^na) == 6i(cosma) = 0,# cos a. Consequently the roots of C) are so connected that every one of them is a rational function of a single one among them, x, and that 0i0(.r) = dt)i(x) (x = cos a) . The equation C) is therefore an Abelian equation. Accordingly 2- x = cos a — cos — // can be algebraically obtained. We have here an example of § 181. § 189. Suppose now that n is an odd prime number, n = 2* -f~ L Then the roots of the equation C) are the following*: 2^ 4r 4v* r0S 27+l' r0 *27+l'--- C0S 2T+l' ' Since the last root is equal to 1, the equation C) is divisible by x — 1. The other roots coincide in pairs 2m~ _ cos (2v -f- 1 — m)2- cos 27+i.~~ ' 2v+i ~* Consequently we can obtain from C) an equation with rational coefficients, the roots of which will be the following 2- 4- 2v7T COS « r~r , cos - — — .... cos ■ 2v + V 2> + l' 2v + l' This equation is of the form 220 THEORY OF SUBSTITUTIONS. CM .«• I,' '+4(,_i) a --»_Kv_2)»' ' + A ^ M" 3 s' ' (y-3)(v-4) _ 1 ■ D With the notation '2- cos = cosa— x 2> -\- 1 we have then 2m- . . cos = ti(x) = cos m a, 80 that the equation CJ has also the roots ), tr(x), ^(a>), . . . that 18 cosa, cosma, cosnra, cosm 3 a, . . .cosmfa, . . . If now g is any primitive root (mod. 2v -f- 1 ) then the v terms of the series -Ri) cosa, cosga, cos (fa, . . . cosg"~* a, are distinct, For from the equation cos g a a — cos g B a (a > ; a, < >) it would follow that or, replacing a bv its value "" , 2v + 1 17 a T ^ = flr* ({? a "^ 1) = fc(2w + 1 ,). Dividing both sides of this equation by gP, and multiplying by ga -p _j_ ^ we ^t a i n the congruence i-»===l (mod. 2> + l). But, since since 2(« — /?) < 2v, this congruence is impossible. Con- sequently cosg a a is different from eosg^a. Again cosg'a — cos equations of degrees n lt n 2 , . . . w w . If n l5 ».., . . . n u are prime to each other, the coefficients of these equations are rational numbers. (§176). In particular if v = 2™, we have the theorem on the construction of regular polygons by the aid of the ruler and compass. J CHAPTER XII. EQUATIONS Willi RATIONAL RELATIONS BETWEEN THREE ROOTS. £ 100. The method employed in £ 1S:> is also applicable to other cases. "We will suppose for example that all the substitutions of a transitive group G are obtained by combination of the two substitu- tions Sj and s 2 , which satisfy the conditions 1 | that the equation sf = sf holds only when both sides are equal to identity, and 2) that .SjS, — .s/.v. (('/. § :]~t). If, then, the orders of s, and s 2 are », and n.,, all the substitutions of G are represented, each once and only once by s,*ia 2 *« (ft, = Q,l,2, ... n. 1). Suppose now that G is the group of an equation f(x) = 0. We construct a resolvent tp — „ + «,<*! + <« 2 V 2 +...<* V„, ,]"■ is therefore unchanged by every *,", and since s a permutes the c *s cyclically, / remains unchanged by all the substitutions of the group, and can be rationally expressed in terms of the coefficients of /(.n. We can therefore obtain <:„. «.',.... by the extraction of an >/,"' root, as in the preceding Chapter. The group of the equation then re- duces to the powers of 8 } , and the equation itself becomes a simplest Abelian equation. 8 191. Again, if a transitive group consists of combinations of three substitutions s, ,s.,,s 3 , for which 1) the equations RATIONAL RELATIONS BETWEEN TIIREE ROOTS. 228 are satisfied only when both sides of each equation are equal to Identity, and 2) the relations hold S& — S^Sjj, 83S] = S^S/Sg, SaSj, = n,°V- s ' . then all the substitutions of the group are represented, each once and only once by Sl V 2 V (fc.- = 0, 1,2. ... n { — 1), where »,, n 2 , //. ; are the orders of Si,s 2 , s 3 . If now G is the group of an equation, we can show by precisely the same method as before, that the equation can be solved algebraically. Obviously we can proceed further in the same direction. That groups actually arise in this way which are not contained among those treated in the last Chapter is apparent from the example on p. 39, where s 2 s, = *,%. § 192. Returning to the example of § 190, we examine more closely the group there given. If we suppose s 2 to be replaced by its reciprocal, it follows from the second condition that s., _, s,.So = Sj fc . From s l we can therefore obtain every possible s., by the method of § 40. We have only to write under every cycle of s, a cycle of 8* of the same order, and to determine the substitution which replaces every element of the upper line by the element immediately below T it. This substitution will be one of the possible s 2 's. We consider separately the two cases 1) where s, consists of two or more cycles, and 2) where .Sj has only one cycle. In the former case the transitivity of the group is secured by s 2 . Consequently every cycle of s, A " must contain some elements different from those of the cycle of Sj under which it is written. It is clear also that all the cycles of s, must contain the same number of ele- ments. Otherwise the elements of the cycles of the same order would furnish a system of intransitivity. The order of the cycles can then obviously be so taken that the elements of the second cycle stand under those of the first, those of the third under those of the second, and so on, so that with a proper notation the following order of correspondence is obtained 22 \ THEORY OJ' SUBSTITUTIONS. S,= {x t X 2 ./• . . . )(//,// //., ...)... • s /' = (Z/i2/i : a/7i (-*• ••)(-!-! fc*i +*•••)••■ It follows then that i n =(37,^2, . . . )(.r,//, ' ; ,.: ...)... The group is therefore non-primitive, the systems of non-primitivity being .<-,. .<-., ...;//,, //_,, . . .; z n .1,,, . . . The substitutions ,s, a leave the several systems unchanged, the substitutions s 1 a s 2 permute the systems cyclically one step, sfsj* two step and so on. Accordingly every substitution of the group except identity affects every element. The group is, in fact, a group !-' (§ 129). The adjunction of any arbitrary element X\ reduces the group to the identical substitution. Consequently all the roots are rational functions of any one among them. The following may serve as an example: S, = l.r,.r,.-- : , (//,//_//.), S 2 = {x //I | '•//) (..//,), s 2 s 1 = (x 1 y 2 ) {x 2 y x ) (x s y 3 ) s £ L93. In the second case, where s, consists of a single cycle, the transitivity is already secured. We may write then, as in Chap- ter VIII, -\Z '2 4- 1 . To construct the s 2 's we proceed as before and obtain from the scries of substitutions s 2 — ; z kz -\-i — k\, s/= z kH - i i k)\. tC — 1 Now. in the hrst place, it is easily shown that the group contains substitutions different from identity, which do not affect all the ele- ments. For among the powers of *, there is certainly one ^, M which has a sequence of two elements in common with s 2 . Then n^.s, ' does not affect all the elements. Again, it can be shown that there is no substitution except iden- tity which leaves the elements unchanged. For we have k? —1 8faf = \z kHz + a) -{---- 1/ /.,, k ~ 1 RATIONAL RELATIONS BETWEEN THREE ROOTS. 225 and if ,t\ and a*A + 1 were not affected by the substitution we should have A-* 3 — 1 #(j + ! + «)+_ ±(i—k)=X + l, and consequently /C" . J. . The substitution then becomes s* s f — \z z + a | , and since .*> and .r A + ] are unchanged, a=0, aDd the substitution is identity. The following is an example of this type: s, = (x 1 x 2 x 3 x i x 5 x & ) , s 2 = {x x x._) (a? 3 a? 6 ) 4 a? 5 ) From the preceding considerations we deduce Theorem I. // the group of an equation is of the kind defined in § 190, all the roots of the equation are rational functions of, at the most, two among them, and the equation is solvable alge- braically. £ 194. We turn now to the converse problem and consider those irreducible equations, the roots of which are rational func- tions of two among them: If any substitution of the group G of such an equation leaves .<■, and .r, unchanged, it must leave every element unchanged. Again, if .s a and s' a are any two substitutions of G which have the same effect on both x t and .r_,, then s' a s a ^ 1 leaves x x and x> un- changed; consequently s' a Sar x = 1, and s' a = s a . Suppose now that the substitutions of G are s 1 , s 2 , s 3 , . . . s,. . There are n(n — 1) different possible ways of replacing :r, and < from the n elements x t ,x 2 , . . . x n . If any one of these ways is not represented in the line above, let t 2 be any substitution which produces the new arrangement. Then the substitutions of the line 15 226 THEORY OF SUBSTITUTIONS. f _..S', , (..N.j. r.N, . . . . I .s will replace a?, and x. by pairs of elements which are all different from one another, and none of which correspond to the first line. If 2r is still that the orders n, and n., can only be 2 and 3, while rijWj must be equal to 12. § 196. If however, the degree of the equation of §194 is a prime number p, we have precisely the case treated at the begin- ning of the Chapter. To show this we observe that by Theorem II, the transitive group Q of the equation is of an order which is a divisor of p(p — I >. Since the transitive group is of degree p, its order is also a multi- ple of p. It contains, therefore, a substitution s { of the p th order, and consequently also a subgroup of the same order. If now in pip — 1\ § 12S, Theorem I, we take a = 1, and put for r the order — -', • RATIONAL RELATIONS BETWBEH THREE BOOTS. 227 it follows that k= 0, that is, G contains no substitutions of order i> except the powers of s, . Consequently we must have 82*1*2 ' = */'• and this is the assumption made in § 188. Equations of this kind were lirst considered by Galois,* and have been called Galois equations. We do not however employ this designation, in order to avoid confusion with the Galois resolvent equations, i. ■ .. those resolvent equations of which every root is a rational function of every other one. If a substitution of the group G of an equation of the pres- ent type is to leave any element .»•_, unchanged, we must have from §193 /J 1 (kfi — l)z + akP + — T (i—k) (mod.//). A' — J_ Since kP — 1 is either I mod. p) or is prime to p, it follows that either every x is unchanged and the substitution is equal to 1, or one element at the most remains unchanged. Theorem III. If all the roots of an irreducible equation of prime degree p are rational functions of two among them, the group of the equation contains, besides the identical substitution, p — 1 substitutions of order p and substitutions ivhich affect p -1 elements. The solution of the equation reduces to that of hvo Abelian equations. £ 197. The simplest example of the equations of this type is furnished by the binomial equation of prime degree p x p —A = in the case where the real p tb root of the absolute value of the real quantity A does not belong to the domain of those quantities which we regard as rational. The roots of this equation, if x r is one of them, are The quotient of any two roots of the equation is therefore a power of the primitive p th root of unity °>x^, <"■> >"■-■> and consequently their quotient ± w T .r/", (m > 0) is rational within the rational domain. Since p is a prime number, it is possible to find an integer //. such that the congruence or the equation ///," = vp-\-] shall be satisfied. Then the quantity (±x 1 '"oj t Y= ±x vp+1 w' lT = ± A"a; 1 a*' 4T == ± A'..'. and consequently ./, is rational. From the reducibility of the e< piation would therefore follow the rationality of a root, which is certainly impossible. The group of the equation is of order p(p — 1). For if we leave one root x t unchanged, any other root <".r, can still be converted into any one of the p — 1 roots iox lt <"'.<,, u?X\ . . . w p ~ 1 X 1 . RATIONAL RELATIONS BETWEEN THREE ROOTS. 229 Theorem IV. The binomial (-(/nation x p —A-0. in which A is not the p tb 'power of any quantity belonging to the rational domain, belongs to the type of § 196. Its group is of order p(p — l). § 198. Remark. By Theorem III every irreducible equation the roots of which are rational functions of two among them is alge- braically solvable. At present we have not the means of proving the converse theorem. It will however be shown in the following Chapter by algebraic considerations, and again at a later period in the treat- ment of solvable equations by the aid of the theory of groups, that every equation of prime degree, which is irreducible and algebrai- cally solvable, is either an equation of the type above considered, or an Abelian equation. Before we pass to such general considera- tions, we treat first another special case, characterized by rational relations among the roots taken three by three. § 199. An equation is said to be of triad character, or it is called briefly a triad equation,* if its roots can be arranged in tri- ads x a , Xp, x y in such a way that any two elements of a triad deter- mine the third element rationally, i. c, if x a and Xp determine x y , Xp and x y determine x a , and x y and x a determine xp . Thus the equations of the third degree are triad equations: for OC^ —j— *X*<2 i~ *%*"£ — ^i • Of the equations of the higher degree, those of the seventh degree may be of triad character. In this case the following distribution of the roots x u x 2 , . . . ,r 7 is possible: Xi , X 2 , X 3 ; X li X ii X 5 ] X l} X 6 ,Xq', .»'_,. .»'_, . .r, : .»•_,, .c . ,r 7 : •^3 5 Xi j X-; ; x 3 , Xr± , x$ . If the degree of an equation is n, there are = — - pairs of roots x a ,xp. With every one of these pairs belongs a third root x y . Every such triad occurs three times, according as we take for the original pair of roots x a , Xp; Xp,x y ; or x y ,x a . There are there- fore ^ — - triads, and since this number must be an integer, it 1 Noether: Math. Ann. XV, p. 8!). 230 ihkiiky OF SUBSTITUTIONS. follows that the triad character is only possible when » = 6m -\- 1 or » = Cym -)-:'> . The case it ~ Vun must be excluded, because ?? must be an odd number, as appears at once if we combine a?, with all the other elements, which must then group themselves in pairs. The general question whether every »i = 6m-f-l, n = 6m + 3 furnishes a triad system we do not here consider. It is however easy to establish processes for deducing from a triad system of )i elements a second triad system of 2w + l elements, and from two triad systems of />, and n ., elements a third system of n } v., elements. From the existence of the triad character for n = 3 follows therefore that for n = 7. L5, 31, ... ; 9, 19, 39 ; 21, 43, . . . These do not however exhaust all possible cases. There are for example triad systems for >i = 13, etc. § 200. We proceed to develop the two processes above men- tioned. In the first place suppose a triad system of n elements .'■, . .<•., . . . x„ given. To these we add ra+1 other elements ./, . .i ■',. .*'.. <•',.. We retain the \n(n - - 1 ) triads of the former ele- ments, and also cpnstruct from these . new triads by accent- iisg in each case every two of the three t's. Finally we form h further triads x , £»,, .»',; o? , .*■',. x\\ . . . , and have then in all \„{n— 1) , (2to + 1)2m —6 f " = 0^ triads, which furnish the system belonging to the 2u -f- 1 elements. For example, suppose n = 3. We obtain then the following sys- tem • .»•, , .r._, , .r.: Xj . '■ . x ;'. x , . Xq . •'■ ; '. x ■ x . ■>','. .'',, . .'■, , x jj which agrees, apart from the mere notation, with the triad system for seven elements established in the preceding Section. § 201. Again, suppose two triad systems of degrees ?/, and //.._, to be given. The indices of the first system we denote by , <-,... , those of the second by a,/?, r, . . . We may designate a triad by the corresponding indices. Suppose that the triads of the first system are RATIONAL RELATIONS BETWEEN THREE ROOTS. 7\) a, b, c; a,d,e; b, d,g; . . . and those of the second 231 T*) "■ p> r; We denote the elements of the combined system by x aa , x a p, x ba , . . . and form for these a triad system as follows. In the first place, we write after every index of T x ) the index a. In this way there arise Hifoi— 1) In the same way we write ft, then y, then », . . . after every index of 7Y). We obtain then in every case —z— - and in all triads of elements with double indices. 6 n 6 triads of the elements x aa , x a p, x ay , . . . x ba , x b p, . . are different from one another. They are aa, ba. ea; da, da, ea; bo, da, go.:., aft, dp, eft; bft, dp, gft; . . ay, dy, ey; by, dy, gy; . . All of these r 3 ) aft, bft, eft ay, by, cy Again, we write every index of the system T x ) before every index of 7\), and obtain n 2 (w 2 — 1) n. 6 triads among the same n,» 2 elements with double indices. These are also different from one another and from those of T' z ) They are T",.) aa, aft, ay; aa, a,8, as; aa, a~, ar,; . . . ba, f> ft . by: bo., bS, />•;; ba, b", by, . . . ca, eft, cy. ea, c<), cs; co., C, Cr, 7> ■ • Finally we combine every triad of T t ) with every triad of T.) by writing after the three indices of a triad of T t ) the three indices of a triad of TV). With any two given triads this can be done in six ways. For example from b, d, g and a , C, ij we have ba, dZ, gr t ; bo. dv t , g^; b-, do, gjj, b~, di l% go.: b r t . da, g£\ br t , dr. ga. 232 THEORY OF SUBSTITUTIONS. We obtain therefore from T,) and T.,) ft w,(n, — 1) ^(^ — 1) _ h,h 2 — ?i, — Ho + 1 0--Q- -g- --Win, g - > such combinations. These are again all different from one another and from those of 2",) and T" 3 ). They are the following: r",) aa, bft, cy\ aa, by, eft a a , b 8 , ce; a«, b s , c« a«, d/9, e^; aa, dy, e/5 aft, ba, cy; . . . ay, bft, ca\ ad, ba, c £ ; . . . a e , bS, ea; a/3, da, ey; . . . ay-, dft, ea; We have therefore now constructed in all Hj(h, — 1) n 2 (n 2 — 1). h,h, — h, — Uo + 1 _ /i|H 2 (n,Ho — 1) H, 5 f- >ij ~~~ ~T~ W1W2 p — rt — different triads among the elements •£«a 5 '"a/3 > ** oy 5 • • • J "X-ba. j Xbp j "Cfcy • • • 5 • • • The three tables T 8 ) therefore form a possible triad system for //,//_. elements. § 202. The triad group for n = 3 demands no special notice. It is simply the symmetric group of the three elements. To determine the group of the triad equation for n = 7 we pro- ceed as follows, restricting ourselves to irreducible equations of this type. With this restriction the resulting group of 7 elements is transi- tive. Its order is therefore divisible by 7, and it consequently con- tains a circular substitution of the 7 th order, which we may assume to be , We determine now conversely the arrangements of the 7 elements in triads, which are not disturbed by the powers of 8, . These must bo such that if x a , Xp, x y form a triad, the same is true for every x a -f- /, Xp -\- i, x y + i (i = 1 , 2 , . . . 6). Again with a proper choice of nota- tion we may take x a = a?, , Xp = x., , since a proper power of 8, will contain the two elements x a , Xp in succession. If now we apply the powers of s, to the system *y* 'Y* / y* • / y* sy* / y* * y* / y* *y* * / y* / y* y* • *• r 1 y* • RATIONAL RELATIONS BETWEEN THREE ROOTS. 233 it appears that only the second and the fourth cases give rise to a triad distribution of the required character, viz. J i ) .r, , .('._, , .1 ,; .('_,, .r, . .»■-,; ,*',, .r 4 , ,r h ; .r 4 , .;■-, Xj] .r- it u* 6 , CC t ; i L i I .i"], x 3i -^'u^ 3?o, a*3, ^ ; ; x 3l x t , .<■, ; .>', , .'■- , ./•. ; .f. , a; ej a? 3 ; The two distributions are not essentially different, each being ob- tained from the other by interchanging x,,.r 7 ; x a , x^ : and x 4 , sc 5 . We may therefore assume that T x ) is given, and that S, belongs to the corresponding group. If there are other substitutions of the 7 th order belonging to the group, a proper power of every one of these will contain a?, and ,»\ in succession. We may write the sub- stitution therefore l.'V'V. •''.,•'•,'•,,•'',;) = (1 2 a 3 o 4 a 5 a, a 7 j To this substitution correspond, as in the case of Sj , only two triad systems, which proceed respectively from 1, 2, a 4 and 1, 2 a 6 . The indices , c^, a 4 ; a 7 , 1, a, are to coincide respectively with 1,2,4; 2.3.5; 3,7,1; 7.6,2; I), 4, 3; 4,5,7: 5,1,6, we must have 'a, = 3, a 4 = 7, a.-, = 6, Og = 4, a 7 = 5, and accord- ingly s = ('.(•,.*•..(• :l .r 7 .i , , i .r 4 .r-,). Similarly we obtain for the seven new s's I »* »• i' »■ m' i' Y* 1 c I -if 'Y* / V* / >* ■>* f »» 1 o I^y* ^Y* ~y* r* >■ o" 'y. 1 2 — V** 1 2 5 4 ; > 7 f i '' ■« — V ^l** 2 li 1' 7' ;;** 5m 4 — \ 1 ' 7 I 6 V 3/J •S- — ( -J'j.r j.l'i l ; .t h .i 4 .f - ;, Sg — ^Cju".>/* -.* ,.< -jX^X^J^ S 7 — [X \Xi>X '^X^X^X fl '-jf , / 'y. y y. /y. ry* /v. />» \ 8 — V*-') _•' 7'' .!•' :,•' 4' G '• Beside the powers of s, , s 2 , . . . s s there can obviously be no other substitutions of the r t th order in the group. We note, without further proof, that it follows from this by the aid of § 76, Theorem XII, that the required group is { S, , So , . . . S s j The same result has been obtained by Kronecker from an entirely different point of view. 234 THEORY OF SUJSTITrTIoNS. Theorem V. The roots of the most gem ral irreducible triad (■(Illation of the 1 th degree can be arranged as follows: 77/r group of the equation is the Kronecker group* of order t68 r defined l>;j : az + b\ t \z aO(z + b) + c (a 1,2. t; b, c±=0, L, . . . 6; 0(«) = - v. f 1)) 7< is doubly transitive. Those of its substitutions which replace' .< . .'•. by .<■;. Xa are I. r.,./', .'M I .'■.'• ,.»-, ). I. (•„.(•,.(•.,) ( .<•,./,.*• |. l.r M .r,.r.) ( .c ,.*' 4 .c,J. (.)'„.*•,.*' ) I -'",.''-,'"„)■ A// Mc.se f//.s-o replace x 3 by jc . Consequent!!/ we have also x = & 1 (x 1 ,x s ), .«-, = -';,(.*•,..*■„), a/(«/ similarly ••-, ''V ■'',•, •<•<). ■'•■ = ''',(-«- 4 . ■'■,>; e/r. A?/ the substitutions of the group which interchange .«•„ and .*•, are <■'•„•<•,) (■'•..■•• '. ''-,•'-,) l-'V'U, (•''„'' i l (-'VVv';); (■'■„•'■,) <• r.,.|-,..-,.r,|. a»'/ since £/iese aM feave .<•. unchanged, it follows that and the same property holds for all the other triads. Every sym- metric function of the roots of a triad is a 1-valued resolvent. § 203. We examine also the triad equations for u = 9. In the construction of the triads it is easily recognized that there is only one possible system, if we disregard the mere numbering of the ele- ments. We can therefore assume the system to be that constructed in §'2()1. and designate the elements accordingly by two indices. c-ach 00,10,20; 01,11,21; 02,12,22: 00,01,02; 10,11,12; 20,21,22: 00,11,22; 01,12,20; 02, 10,21; 00,12,21; 01,10,22: 02.11.20: A characteristic property of every such triad is the condition i"t- i>' ~f" bq + "■. "'/' -\~b'q -{-<*■'; ap' +67' fa, a'p' + b'q' -fa'; ap" + og" + a, ay + 6V + «"; and if the condition B) is satisfied by P,p',p"',q,q',q", it is also satisfied by the new indices. Conversely, every substitution that leaves the triad system un- changed can be written in the form s by a proper choice of the coefficients a, 0, a; a', //, a'. For if f 1 , is any substitution of the triad groiq> which replaces the index (0, . b' '. Consequently t 3 = t.,*.r ' will leave both (0, 0) and (0, 1 ) unchanged. Again if / :; replaces (1, 0) by (c, <•'), then s 3 = p,q cp, c'p + q will leave (0, 0) and (0,1) unchanged, and will replace (1, 0) by ice') consequently t i — f..s : . '. which belongs to the group of the triad equation, will leave (0, 0), (0, 1) and (1, 0) unchanged. A glance at the triad system shows that we must have t t == 1, and it follows accord ingly that Consequently t, is actually of the assigned form. Remembering further that we have established in § 145 the necessary and sufficient condition that this form shall actually furnish a substitution, we have the following 236 THEORY OF SUBSTITUTIONS. Theorem VI. The group G of the irreducible triad equa- tion of degree •'. consists of all the substitutions S }>. ij i ip -|- bq -f- a , a p -\- b'q -\- a' (mod. 3 ) ab' '('!> — (mod. 3) Tlic order of G is, from £ 145 r = 3 8 (3 a 1 ) {:{-' — :i) = 27 - 10 The root* of the equation are connected, in accordance with the triad system, as follows: — ''{■'': ''in). -''21 —- '''•''oil ■'ll'" •'.'.' = ''(-''hj- -''i.'); • • • All the substitutions of G which replace .<•„„ and .»•,„ by x Vl and ./•,„ are of the form s '= P> 2 p-\~bq-\-l,b'q (mod. 3), and since these all concert .<•_,,, into .*■„,,, it follows that ice Juice also •''mi ''(-''lIM •''•.>(|)j ^10 = ''(-''.IK - ' 'lM. ) - ■ • • All the substitutions of the group which interchange x M and .r w are of the form s" = p,q 2p + bq + l,b'q (mod. 3), and since these all leace .r^ unchanged, ice have, again, •'\," ''/(.r 1MI , •'•„,) = >'H-c w , •'',...); • • • § 204. The arrangement in triads given at the beginning of the preceding Section possesses a peculiarity, which we can turn to account. The triad system is so distributed in four lines that the three triads of every line contain all the ( .) elements. Evidently every substitution of the group permutes the several lines as entities among themselves. We determine now those sub- stitutions which convert every line into itself. If a = />, 1 1 a j> -)- bq -f a . a'j> + b'q -\- «' ( mod. 3) is to convert the first line into itself, the new value of 7 must depend solely on the old value of q, but not on the value of p. Consequently we must have a' = 0. If the substitution is to convert the second line also into itself, we must again for the same reason have t> = 0. The substitution is therefore of the form <- />.2 2 , since ab' - (mod. 3). It is further required that a shall also leave the third and fourth lines unchanged. The third line has the property thatfin every triad (P ( h P f< l'iP" l l") the three sums +' I ' ' t I ft P + '/ have respectively the values 0, 1, 2 (mod. 3); the fourth that p + q= p' -f q =p" + q" (mod. 3). If now we apply a to the triad (00, 12, 21) of the fourth line, we obtain (a, a'; a + «, 26'+ a'; 2a + «, 6'+ a') (mod. 3), and consequently we must have u + a! =a+ 2b' + « + «' : 2a + b'+ a + a' (mod. 3), that is, a b' (mod. 3). The tinal form of , a — 1. ami we must take "= P? Q P + "> 7 ■ The r's form again a self-conjugate subgroup of I of II oi ordei '■>. We construct by §86 the quotient U = H:I. V is of order 3-2 and of degree 3, corresponding to the three triads. U is therefore the symmetric group of three elements. If. then, we construct a function O of the 9 elements x, which belongs to the group /. this latter adjunction of '•> ',..-..,. and •'■',. .>■',. .»■'., have two roots in common, then they have also the third root in common. For. if .?■, — .»•',, ./ .• ■'. . it follows that and if x\ and x 3 are not the same root, the given equation, having equal roots, would be reducible. If x t is a root different from .<■,. .,■,. .v.. there is a substitution in the group which replaces .r, by .r 4 . If this substitution leaves no element unchanged, we obtain an entirely new system ■'•,,■'•-,,'•,,. But if one element, for example .v.., remains unchanged, we have for anew system .»•_., .<■,, .c 7 . Proceeding in this way, and examining the possible effects of the substitutions, it is seen that all the roots arrange themselves in the triad system of 9 elements. Comparing this result with Theorem VI, it appears that the equation is exactly one of the triad equations just treated. It is known * that the nine points of inflection of a plane curve of the third order lie by threes on straight lines. These lines are twelve in number, and four of them pass through every point of inflection. Any two of the nine points determine a third one. so that the points form a triad system, as considered above. The abscissas or the ordinates of the nine points therefore satisfy a triad equa- tion of the 9th degree, and this equation, belonging to the type above discussed, is algebraically solvable. It can, in fact.be shown that if -<\, .>-,. .<■ , are the abscissas or the ordinates of three points of iuilection lying on the same straight line, then .r =0{ »-,. x 2 ), x s = 'H-c,. .r,), x 2 = "(.<■, a?,), where is a rational and symmetric function of its two elements. The discussion of this matter belongs however to other mathemati- cal theories and must be omitted here. *0. Hesse: Crelle XXVIII, p. 68; XXXIV, p. 191. Salmon: Crelle XXXIX. p. 365. CIIAPTKU XIII THE ALGEBRAIC SOLUTION OF EQUATIONS. § 207. In the last three Chapters various equations have been treated for which certain relations among the roots were d priori specified, and which in consequence admitted the application of the theory of substitutions. In general questions of this character, however, a doubt presents itself which, as we have already pointed out, must be disposed of first of all, if the application of the theory of substitutions to gen- eral algebraic questions is to be admissible. The theory of substi- tutions deals exclusively with rational functions of the roots of equations. If therefore in the algebraic solution of algebraic equa- tions irrational functions of the roots occur, we enter upon a re- gion in which even the idea of a substitution fails. The funda- mental question thus raised can of course only be settled by alge- braic means; the application to it of the theory of substitutions would beg the question. To cite a single special example, proof of the impossibility of an algebraic solution of general equations above the fourth degree can never be obtained from the theory of substitutions alone. §208. In the discussion of algebraic questions it is essential first of all to define the territory the quantities lying within which are to be regarded as rational. We adopt the definition* that all rational functions with integral coefficients of certain quantities ))\'. S M". St"', . . . constitute the rational domain (9t, 9t" '.K'". . . . ). If among any functions of this domain the operations of addition, subtraction, multiplication, divis- ion, and involution to an integral power are performed, the result- ing quantities still belong to the same rational domain. The extraction of roots on the other hand will in general lead *],. Kroncckrr: I5erl. Ber 187*, !>■ 205 II.: Cf. also: a r: t h in. Theorie d. algob. Grossen. THE ALGEBRAIC SOLUTION 01" EQUATIONS. 241 to quantities which He outside the rational domain. We may limit ourselves to the extraction of roots of prime order, since an (mn) a> root can be replaced by an j;/" 1 root of an u lh root. All those functions of 9t', 9t", 9t'", . . . which can be obtained from the rational functions of the domain by the extraction of a single root or of any finite number of roots are designated, collect- ively, as the algebraic /mictions of the domain (91', 31", 9t'", . . .). In proceeding from the rational to the algebraic functions of the domain, the tirst step therefore consists in extracting a root of prime order p v ol a rational, integral or fractional function ^,,(91', 9t", 9t'" . • .) which in the domain ( s .)i\ 9t'', 91'", . . .) is not a peifect p u ib power. Suppose the quantity thus obtained to be V v so that F?" = * T v (9r,9r,9T',...). We will now extend the rational domain by adding or adjoining to it the quantity V v , so that we have from now on for the rational domain (V„; 9t', 9t", 9ft"', . . .), i. e., all rational, integral or fractional functions of V„, 9i', 91", W", . . . are regarded as rational. The present domain includes the previous one. With this extension goes a like extension of the property of reducibility. Thus the function x p — F„CSt', 9i", . . .) was originally irreducible: it has now become reducible and has, in the extended domain (V„; W, 9t", . . . ). the ra- tional factor x — l', .. The new domain can be extended again by the extraction of a second root of prime order. We construct any rational function which is not a perfect (p v _ v th power within (TV, 5Jt', Jft", . . ., and denote its (p,._i) lh root by F„_ n so that It is not essential here that V v should occur in F\,_ x If now we adjoin V v _ x , we obtain the further extended rational domain (F„_,, V v \ 91', 9t", . . .). Similarly we construct V p »_- 2 *=F v _ 2 (V v _„V v ;W,*t",...), F r L7 8 = f,_,(1V_ tl V V _ X1 V V \ 91', 91", . . .), V p > = F 1 {V 2 ,V a ,...V v ;W, %",...), 16 THEORY OF SUBSTITUTIO. where the P's denote rational function- of the quantities in paren theses, and Pi,jo 2 , . . ■}>,-: are prime numbers. Any given algebraic expression can therefore be represented in conformity with the preceding scheme, by treatin ame way in which the calculation of such an expression involving only numerical quantities is accomplished. £ '200. Tho FaS are readily reduced to a form in which they are integral in the corresponding V"s. that is V a+l , V a+2 . . . V v , and are fractional only in the 9ft', 9ft", . . . Thus, suppose that where 6r , G x , G 2 , . . . ; H„, H } , H 2 , . . . are rational in V a+2 , F + 3 ,... I \: 9ft', 9ft", . . If now co is a primitive d>o+i) ,h root of unity, the product Pa + 1 - ' A = is a rational function of V a+2 , V a + 3 , . . . V v \ s Jt\ 9t", . . . For on the one hand the product is rational in the H's, and on the other it is integral and symmetric in the roots of VZtf=F.+x{V*+» . . . V v ; SR', 9ft", . . .) and is therefore rational in the coefficient F a f] of this equation. Again, if we omit from the product P) the factor H Q -\- H x V a + x + H 2 V 2 a + l + . . . , the resulting product Pa + 1 - ' P,) JJ[H + H l **Vl +2 + . . .] A=> is integral in V a + i and rational in F a+2 , . . . V v ; 9t', 9t", . . . More- over, since to does not occur in P) or in the omitted factor, it does not occur in P,). If now we multiply numerator and denominator of F a by Pj), the resulting denominator is a rational function of T^a + 25 • • • V v 'i 9t', 9t", . . . alone, while the numerator is rational in these quan- THE ALGEBTUIC SOLUTION OF EQUATIONS. 243 tities and in V a +i* Dividing the several terms of the numerator by the denominator, we have for the reduced form of F a F a = J + J l V a + l + J,Vl + l + .... where the coefficients J ,J l ,J 2l ■ ■ ■ are all rational functions of J\i+2j • • • V v ' t 9ft', S W, ... On account of the equations K\V=Fa + u v p a %y +l = F a+1 v a+} , . . . we may assume that the reduced form of F a contains no higher power of V a + l than the (p a ±\ — I / u - The several coefficients J can now be reduced in the same way as F a above. By multiplying numerator and denominator of their fractional forms by proper factors, all the J's can be converted into integral functions of V a+i of a degree not exceeding p a+2 — 1, and with coefficients which are rational in V a+3 , . . . V v ; 9i', 9ft". ... In this way we can continue to the end. § 2 10. We have now at the outset to establish a preliminary theorem* which will be of repeated application in the investigation of the algebraic form peculiar to the roots of solvable equations.* Theorem I. If /„,/,, . ..f P -i',F are functions within a definite rational domain, the simultaneous existence of the fico equa- tions A) /o+/^+/.«' 2 + • • • -i/,-,^- 1 =0, B) w*—F =0, requires either that one of the roots of B) belongs to the same rational domain with / ,/n . . • f,,-i', F, or that /o = 0, f l = 0,...f p _ 1 = 0. If all the /„./,. . . ./,,_] are not equal to 0, the equations .1) and B) have at least one root w in common. In the greatest common divisor of the polynomials A) and B) the coefficient of the highest power of w is unity, from the form of B). Suppose the greatest common divisor to be C) cr 4- (.',?(' -f- c,V- -+-....+ IC V . Equated to 0, this furnishes v roots of B). If one of these is deno- *Tliis theorem .vis urighiail) i^iveii by And: Deiivre-* coui|ji&ieH II, luii. Ivrou- euker was Hie Qrst lo establish it in the lull importance: Berl. Her. 1879, p. 244 THEORY OF SUBSTITUTIONS. ted by IP,, and a primitive p tb ro t of unity by w, then all the - roots of C) can be expressed by U'l, io a U\, tfPll\, fttfJC,, . . . Apart from its algebraic sign, tr is the product of these roots Now since p is a prime number, it is possible to find two numbers u and v. for which pu + v v = 1 , and consequently (± toY=a>*w l 1 -'; ,./ & ic ] = F"(±Y f y. One root, a, r *«j,, of the equation i?) therefore belongs to the given rational domain. § 211. We apply Theorem I first to the further reduction of If J K is anyone of the coefficients ./../.. . which does i^pt vanish, we determine a new quantity W a+ ] by the equation A,) W a+l ~J K Y 0, annex to thi9 the equation of definition for V&+ , and fix for the rational domain R) (T^. + ,;F , . + „V. + „... ;»',»",...), It follows then, if A) and i?) of Theorem I are replaced by .1,) and £,), that, since the possibility TF. +1 - 0, J, = is excluded, we must " hive C,) «r. +1 = i2(W. +1 ; r a F,; «',»", ...» where w is a (p„ I th root of unity. We can therefore introduce into the expression for /•'„ in the place of V a + l the function W a ,. provided we adjoin the (p a ,) lh root of unity, <", to the rational domain. From .1,) and U . i ii is clear thai (W.+,! F. +t ,...! tt'.ffi", ...ina-lll'..,.!', , • . ". ; *',»",...). define the same rational domain, and tho equation B') w?£ l = J»«+- F25+ l ==#. +1 - 1 , where G , G iy G s , . .. are integral functions of V 2 , V 3 , . . . V„ and ra- tional functions of 9ft', W, '. . . , and G x may be assumed to be 1(§ 211). Taking the powers of x and reducing in every case those pow ers of F, above the (p, — l) u ', we obtain for every v If these powers of x are substituted in 1), we have A) f(x ) = H + H 1 V 1 + H. 2 V< + . . . +Jff J1 _ l FV\ where the iTs are formed additivily from the G (,, s and the coeffi- cients of 1). Joining with A) the equation of definition of V x 240 THEORi uh SUBSTITUTIONS. and applying Theorem I to A) and B), we have only two possibili- : cither a root of B) is rational in the domain ( V 2 , V 3 , . . . V ; 3t', »", . . . ), or Ho = 0, ff, = 0, ff f = 0, ...#,„_, =0. Both cases actually occur. In the former the scheme 2), by which we passed from the original rational domain to the root x Q , can be simplified by merely suppressing the equation Vfi = F l (V 2 ,V 8 ,...), and adding the ^, th root of unity to the rational domain. § 213. As an example of this case we may take the equation of the third degree f(x) = x* — Sax — 26 = 0, the rational domain being formed from the coefficients a and 6. By Cardan's formula •''..= N / 6 + ] + V b — y 6* — a 3 - This algebraic expression can be arranged schematically as follows: V 9 2 = b*—a\ V,? = b+V z , Vl 3 = b—V 3 . --v*+v\. The expression for ' j | c ), formed as in the preceding Section, then becomes y(x ) a y 2 +(F 2 a — <*)7i+V a T7 = 0. Comparing this with F, 3 — (6— V a ) = 0, and determining \\ from the hist two equations, we obtain Q so that Vj is already contained in the rational domain ( V 2 , V :i ;a, b). If we now transform F, into an integral function of V. 2 by the pro ceio of § 20'J. we obtain from the relations TEE ALGEEBAIC SOLUTION OF EQUATIONS. 247 [a 2 + (6 + V a )wV a — tWFj, 2 ] [a 2 + (6 + T 3 ) W 2 F 2 — a^F, 2 ] = 2b(b+V,)(a+V.?), [a 2 +(6+ ^)V 2 -aF 2 2 ][26(6 + F 8 )(a + F/)]=[26(6 + V 3 )]\ [a(6 + y 3 ) — a" F 2 + (6 — F 8 ) TV] [26(6 + V 3 ) (a + y 2 2 )] = ±ab\b+V 3 )Y . where wisa primitive cube root of unity, the simpler form _ 4»6 2 (6+TV)Tv _ aV, 2 l ~ = 46 2 (6+F 3 )"' ~b+V t ' Removing V 3 from the denominator by multiplying both terms of the fraction by b — T' ; , we have finally and herewith the reduced form of V x can therefore be suppressed in the scheme above. § 214. We return now to the results of § 212 and examine the second possible case. In f(x ) = H + H l V l +H 2 V 2 + • • • +H Pl _ 1 V^~ 1 suppose that V x is not rational in the domain (V 2 , V„, . . .' ; 9ft', Ot", . . .). Then from Theorem I - #„ = (), H,=0, H 2 = 0,...H in _ 1 =0. If now, in analogy to 3), we form the expressions 3') x k = G + GrfVi + ft-» a F«+ . . . G^o^-^V'r', (fc = 0,l,...p,— 1) in which a> 1 is a primitive p, th root of unity, it follows, with the same notation and process as in § 212, that x\ = G[ v) + GfVT, + g?«*v? +..., f{x k ) = H + H^ V, -f H^V* + . . . Since the if 's vanish identically, the latter expression is also equal to 0, i. e., x k is a root of f(jc) = for k =1,2,... Pi — I. For example, in the case of the equations of the third degree 248 THEORY OF SUBSTITUTIONS. .r 3 — Sax — 26 = 0. where the first of the two possibilities above has been excluded bj reducing x to the form the other two roots are x*=Vi + b —^V,\ ex x a = a?V t + -,-*«> V, (.-=!¥=■) b— V, a 2 § 215. If now we make the allowable assumption (§ 211) that G l = 1, (whereupon F, may possibly take a new form different from its original one), we obtain by linear combination of the p, equa- tions for jr , a;, , . . . x n _ , x =G +V l + G 2 V l *+ . . . + fl^.jTV*- 1 , f. x l = Qo + "V 1 + G,» l *V*+ . . . + GL_ l «*- 1 TVi-\ *«-i = G [ o + ^~ , T 7 i + G> 2 to- 1 >F l 2 + . . . + 6? M _ 1 o»CPi-») 2 v i i»,- the value of V, : i ■ v\ - i r,_ >,2 F, — — > to. k Xv fc=„ The irrational function F, of the coefficients is therefore a rational and, in fact, a linear function of the roots x ,x n . . . cr ;i _, as soon as the primitive p, th root of unity to, is adjoined to the rational domain. § 210. In the construction of the scheme 2) it is not intended to assert that F a necessarily contains V a _ ] , V a -2, ■ ■ . If V a _ t is missing in F a , another arrangement of 2) is possible; we can replace the order V P a %V = F«+i(V a+a ,...), V* a * = F a (V a + 2 ,...), V'Xr = K{V a ,...) by the order v r a - = F a+] (v a+i , . . .), v*fi (v a+2 , . . .), y;.-' = F a _ 1 (v a , . . .). It is therefore possible, for example, that different Vs occur at the end of the series 2). In this case different constructions 3) for the THE ALGEBRAIC SOLUTION OF EQUATIONS. 248 root .i„ are possible, and the theorem proved in the preceding Sec- tion holds for the last V of 2) in every case. To prove the same theorem for all V's which occur, not in the last, but in the next to the last place in 2). we will simply assume that F } actually contains V... The proof (§215) of the theorem for r ; was based on the fact that an expression G» + V y + G 2 V{+ ... satisfied an equation with rational coefficients. We demonstrate the same property for an expression L»+V 2 + L 2 V.{+ ... If we suppose all the permutations of the roots of the equation 1 i to be performed on His— r: fc = the product of the resulting expressions is an integral function of //, with coefficients which are symmetric in the sc's and are therefore rational functions of 5R', 9i", . . . If we denote this function by ;,. .r, c„ ,; conse quently V, is a rational integral function of the p, th degree of the roots of /(.r) — 0, provided the quantities w, and <», are adjoined to the rational domain. In the same way every V can be treated which occurs in the next to the last but not in the last place in 2). Proceeding upward in the series we have finally. Theorem IT. The explicit algebraic function x , which sat isjies a solvable equation f(x) = 0, can be expressed as a rational integral /miction of quantities v lt v a ,v t ,... v v , urith^ coefficients which are rational functions of the quantifies ft', ft". The quantities V\ are on the one hand rational integral functions of the roots of the equation f(x) = and of primiti -e roots of unity, and on the other hand they are determined by a series of equations V a ^ = F(V a _ 1 , V a _ 2 , ...V v - W, ft", . . . ). In these equations the p x ,p 2 ,p 3 , . . . }>„ arc prime numbers, and F x , F., , . . . F v are rational integral functions of their elements V and rational junctions of the quantities ft', ft", . . ., ichich detenu ine the rational domain. § 217. This theorem ensures the possibility of the application of the theory of substitutions to investigation of the solution of equations. It furnishes further the proof of the fundamental prop- osition : Theorem III. The general equations of degree higher than the fourth are not algebraically solvable. For if the n quantities .»•, . .,-. r„. which in the case of the general equation are independent of one another, could be algebra- ically expressed in terms of ft', ft", . . . , then the first introduced irrational function of the coefficients, V v . would be the p v th root of a rational function of ft', ft", . . . Since, from Theorem II, V v is a rational function of the roots.it appears that \\ . as a /'..valued func- tion of .<•,. .<\ . . '„, the p v th power of which is symmetric, is either the square root of the discriminant, or differs from the latter only by THE ALGEBRAIC SOLUTION OF EQUATIONS. 251 a symmetric factor. Consequently we must have p v = 2 (§ 56). If we adjoin the function V,, = .S, \f J to the rational domain, the latter then includes all the one-valued and two-valued functions of the roots. If we are to proceed further with the solution, as is nec- essary if n > 2, there must be a rational function F„_, of the roots, which is (2p v ,) valued, and of which the(p„_i) th power is two-val- ued. But such a function does not exist if u > 4 (§ 58). Conse- quently the process, which should have led to the roots, cannot be continued further. The general equation of a degree above the fourth therefore cannot be algebraicallv solved. § 218. We return now to the form of the roots of solvable alge- braic equations 3) x = 6? + V y + a,T7+ • • • + G Pl _,V^-\ We adjoin to the rational domain the primitive p x n \ p 2 th , . . . roots of unity, and assume that the scheme which leads to x is reduced as far as possible, so that for instance V a is not already contained in the rational domain (V a _ 1 . . . V v \ 9t', 9t", . . . ; <" n <» 2 , . . ♦). We have seen that the substitution of wfVy (k=l t 2,... Pl -l) for V in 3) produces again a root of f(x) = 0. We proceed to prove the generalized theorem: Theorem IV. If injhe scheme 2), which leads to the expres- sion 3) for ,'■„, any V a is multiplied by any root of unity, the values V a -i, V, • ■ ■ V 2 , Vi trill in general be converted into new quan- tifies r a . r a ._,. . . . r.,,i\. If the latter are substituted in the place of the former in the expression for x , the result is again a root of f{x) = 0. We may, without loss of generality, assume that /(.<•) is irredu- cible in the domain QR', St", . . . ). Starting now from 3), and denoting by <" T a primitive r ,h root of unity, we construct i'\ — i /i - 1 JJ(x—x k y=TJ[x—{G + a,*V 1 +G 2 2) is the lowest V that actually occurs. Then 1> JJ(x—Xi)=f a (x; l'„, V a+1 ,...) = a +a 1 V a + a i V a *+ ... The «'s which occur here belong to the domain ( V a .,,...). We construct further : " /^/:,(. l ' k V l i + . . .)] A = o will only differ from those obtained above by the introduction of the gr's and y'a in place of the G"s and Vs. since in all the reductions 2' ) replaces 2). Consequently this product is equal to/„(,r; v„, r„ + M . . . ) and similarly »„ - 1 ///■(»; '"./''«< t' B + n . . .)=/,,(.<': r,,. r„ ,....), A=0 -/'ft— 1 ///•(*; 5 "V ''/, 5 ''/,-.- |i ••• ) /.*•'"• '', 1 V C+1) • ■ • )) A = u l>, — 1 JJf c (x, o^r.r,- „...) = /«(«; »', »",..- )=/(*)■ A = 111 This furnishes the proof that !„ is a root of /(as) = 0. We have still to prove the irreducibility of /„(•<•>, //,(•«), . the rational domains (V„, V a+1 , ...),( T,.. V h .,....)..., respect- ively. Assuming the irreducibility of /„(as) in the domain ( I ','. V„ .,,...). we proceed to demonstrate that of /,,(.<) in the domain ( V b , V,, , , , . . .). The method employed applies in general. If cr(,r; V b , . . .) is one of the irreducible factors of fi(.r), so chosen that it contains /„(as; V ) as a factor, then we have in the domain V„ , V„ + , , . . . the equation 8) .;' I ' Again. /„(.<•; w« V ) is different from fj.r; u>%V„, . . .). For if we write f a (x\ ]'„....) — e + e, V„ -f ^ IV + it would follow from the equality of the two functions /„ that A,) Bl (o>.*- <) F a + £2 K 2 -— ^)y 2 + ...=(), and consequently, from the equation of definition B,) F*— FJLV m+l ,...) = Q, that T a must be rational in the domain (V„ , , , . . . 31', . . . »,, . . .), since a=j=/3. Accordingly /„(.<'; F„, . . .), /„(.*•; w„F„, ...),.. . are all divisors of (f. All these functions are different from one another, and they are all irreducible in the domain (V„, V a+} , . . . ). Consequently y con- tains their product, which, on account of the degrees of /„ and f b in ./•. is possible only if c and/,, coincide. Since the foregoing proof holds for every irreducible factor of 1 ). it still holds if we drop the assumption of irreducibility. $ 219. At the beginning of the preceding Section we remarked that in the product construction with V x other I 's might vanish. This possibility is however excluded in the case of certain l'"s. as we shall now show. We designate any \\ of 2) as an external radical when the fol- lowing Vt^Y r T im • ■ ■ '• ''■< l'\ ii l'\ • do not contain \\. Every such external radical can be brought to the last, position of 2), and the expression of .<■■„, as given in 3), can be arranged in terms of every external radical present. We shall see that in the product construction with \\ no other external radical can be missing. Thus, if l' r is missing in pi - 1 /.,<■<■: i'.., . . .) = JJ[x — (g + «>i k Vi + o>v A iv+ •■•)!, \-0 THE ALGEBRAIC SOLUTION OF EQUATIONS. 255 then /„ cannot be changed if we replace l' r in the fundamental radical expression by <» T K V T , without thereby changing l',. If, as a result, the G'a are converted into the y's. we should then have also ./;,(.<•: r,,...i = //l.^fe+», A F, + !/ ; ^r-...)|. Every linear factor in x of this last expression must therefore be equal to some factor of the preceding expression A) g +V 1 +g 2 V* + ...= Qo + ^Vj + 0, «,*!?+ . . . Taking into account the equation of definition B) V»i-F 1 (V. 2 ,...) = Q, it follows from Theorem I that either V } is rational in V. 2 ,V :] , . . . u> T , which may be excluded, since otherwise 2) could be reduced fur- ther on the adjunction of w T , or that 9o — Go, 9-2 = G-2, ■ ■ ■ In some one of these equations V T must actually occur. Develop- ing this equation according to powers of V T , we have A,) K + K 1 V T + K i V T 2 + . . . =K ll + K,oj r V r + K 2 co r ' i V T 2 + .... and combining with this B t ) V*r—F T (V T+1 ,...) = 0, the impossibility of both alternatives of Theorem I appears at once. Consequently V T could not have been missing in the product con- struction. If we consider only /„ (§ 218, 4) ), the series 2) ending with V„ can also contain external radicals, in fact possibly such as are not external in respect to the entire series. These also cannot vanish in the further product construction. The irreducibility of /„ being borne in mind, the proof is exactly the same as the preceding. Theorem V. In the product construction of the preceding Section no external radicals can disappear from f„ except T',. The same is true for f,, in respect to the external radicals occurring among V V ,V V+1 , . . . V„, and so on. If several external radicals occur in x or in one of the ex- pressions f a , //,,/<-,.., the product of all the corresponding expo- nents is a factor of n. 25* ') THEORY OF SUBSTITUTIONS. Theorem VI. If an irreducible equation of prime degree it is algebraically solvable, the solution will contain only one exter- nal radical. The index of the latter is equal to p, and if w is a primitive p th root of unity^ the polynomial of the equation is r 1 • f(x) = //|< {G +u> K V 1 + G ? a> 2K Vi i +...+G p ,«<* "HY ')]. A = Theorem VI T. If the algebraic expression 3) x = G + T, + G, V{ +... G Jn _ x V* - 1 is a root of an equation f(x) = 0, which is irreducible in the domain (9T, 9t", . . . ), and if we construct the product of the p, factors, in which \\ is replaced by <", 1',. "'f'F, . . . /„(.»■: l r „, . . .)=JJ(x- ■ i. where \' is the lowest V present, and again the product f b (x; V,,,...) of the ji„ factors /„(.r; w„ A F„, . . . ), and so on, we come finally to the etjuation f(x) — 0. the degree of which is n —ViPaPh ■ ■ • The functions f, ,f ,,,... are irreducible in the domains ( V„ , V„ + , , . . .), {v b , r h+1 , ...),. .. § 220. We examine now further those radicals which vanish in the first product construction. The remaining V„ , V„ .;.,,... are not altered in the product construction. We may therefore add these to the rational domain, or, in other words, we may consider an irreducible equation f(x)=fj->: V . . . . ) in the rational domain (V a ,V ;&',&", ..... Here all the V u V,, ... V„ . , already vanish in the first product construction. We examine now what is the result of assigning to V„_, any arbitrary value consistent with its equation of definition, then with this basis assigning any arbitrary value of V„ ., consistent with its equation of definition, and so on. Suppose that the functions I .. i > ^a-2J • • • V'li ^i> G , G 2 , • • • G p ^i are thereupon converted into THE ALGEBRAIC SOLUTION OF EQUATIONS. 257 Va — i » v o— a » • • • v i t v i 5 0o 5 Qi i ■ • ■ 9p — i • The new value assumed by x is then Co = 0o + fir^'i + SW 2 + 9W 3 + • • • + 9 P - i*'i p ~ '■ From § 218 ?„ is again a root, and this together with the system ? n l 2 . . .. fp_i, which arises from c when u, is replaced by a>v x , o> 2 f], . . . . Also p G is the sum of all the roots, and is therefore a rational function in the domain (9t', 9i", . . .). Again we obtain from the system above the equation p Vl =G (l+ oi- 1 +»-»+ . . .) + F(V + a>»v- x + u>'"a>-*+ . . .) + ... Here the first term on the right vanishes. We denote the paren- theses in the following terms briefly by p&uP&a p-a, • ■ • , and write 9) 4 = A, F, + Q,G 2 V* + 9. Z G Z F, 3 + . . . On raising this to the p th power A) vf = Ffa,tk, . . . jBU-ii W, • • .) = [ fl J 7 i + *WW+ • • •]* = A + A 1 V 1 + A a V 2 a +..., and annexing the equation of definition B) Vf = ^(F,, F 3 , . . . F„_,; 3t', . . . ), it follows from Theorem I that either V x is rational in F 2 ,F 3 ,... F _,; u 2 ,v 3 ,...u -i; 8f, »",..., or that F^ = ^ , A, = 0, ^, = 0, ... J,_, = 0. 17 258 THEORY OF SUBSTITUTIONS. We consider now the first of these alternatives. In the rational expression of V t in terms of V 2 , V s , . . . ; r_,. /,, . . . ; SR' 9t", ... all the v 2i v 3 ,. . ,v a _i cannot vanish; otherwise V., should have been sup- pressed in 2). If then we define V 1} as in §§ 208 and 212 by a system of successive radicals, some V K will occur last among the u's and some V\ last among the u's. If we substitute the expression for F, in x , we have x> = B(V l ,:. . V a +»W,. . .) = A(r 2 , . . . V a _ liVi , . . . »._,;»', . . .) Here all the v's cannot vanish, as we have just seen. For the same reason all the 7's cannot vanish, since we might have started out from l . But V K and i\ are two external radicals, and the product of their exponents must therefore be a factor of p (Theorem V). This being impossible, the first alternative is excluded. Accordingly we must have in A) TV = A , A, = 0, A 2 = 0, . v . A p _, - 0. The question now arises what the form of 9) must be in order that its p th power may take the form Vf = A . The equation A) is vf= A + A l V l + AJT? + . . . = [fl, F, + flt^ V," +.--Y- The result just obtained shows that the left member is unchanged if V x is replaced by a> V x , to 1 V x 2 , . . . Consequently [o i u>V l + LLG 2 orV:+...Y = v>; [uyv i + 2 G 2 = 0,...L> K _ ] G K _ 1 = Q, Q K+1 G K+1 = 0,..., that is, 9) reduces to the single term 9') Pi = C«G«TV. THE ALGEBRAIC SOLUTION OF EQUATIONS. 259 Substituting this result, together with g = G in the expression for f , we have c = G + £i K G K V 1 * + g 2 (a K G K V l ' 10) v l = K + 2 » • • • -**JC + p — 2 5 and, if the same operation is performed « times, I) is replaced by -L*a.Ki -^a»c-|-l) -^aK + 2» • • • +-^aK + ]> — 2 J where the indices are of course to be reduced (mod. p — 1). If there is another modification of the radicals, which converts R into i? M , this on being repeated ,'i times converts the series I) into * -R/3M) -fiW + 1? -fiW + 25 • • • -fiW+2>-2« Finally if we apply the first operation a times and the second /S times, I) becomes RaK + P/Jii ■£»aic + Pn + n Ran + Ptn + k—l- Here a and /9 can be so chosen that ax-^-fS/x gives the greatest common divisor of /■ and //. Consequently if R,, is the i? of lowest index which is obtainable from R„ by alteration of the radicals, every other R obtainable from R in this way will have for its index a multiple of k, so that the permutations of the .R's take place only within the systems Ro, ■#/.-> ^2* • • • R n>-i A,. Ri > -Ra -f 1 ) i2 2i + , , . . . R(lZzzJ _ i) /, + ! Here k is a divisor of p — 1. There are then alterations in the meaning of the radicals which produce the substitution (R„ R k li,,, ...)(/?, /.',. | Bj, . , ...)... § 2'22. The preceding developments enable us to determine the group of the irreducible solvable equations 1) of prime degree p. Every permutation of the .r's can only be produced by the alter- THE ALGEBRAIC SOLUTION OF EQUATIONS. 261 ations in the radicals V 11 V 2 , . . . F a _,, and consequently only such permutations of the a*'s can occur in the group as are produced by alterations of the F's. From the result of the preceding Section Vj can be converted into a> T G ^.F,' , and the possible alterations in F, do not change this form. Substituting this in the table of § 215, we have r =U +^G>F/"-+..., ,- ] = G + ^ + 1 G f ,F/+..., We examine now whether any root x^ can remain unchanged in this transformation. In that case -we must have 0.+ . . .+w*Qj.Vt+... = G + *r+*G/>Vt + and from the method which we have repeatedly employed it follows, as a necessary and sufficient condition, that !>e = fi -f- r (mod. p). If e fc =l (mod. p), then for r==0 there is no solution //., and therefore no root or^ which remains unchanged. But for r = 0, every 2 > 9u • • ■ are rational functions of V x p in the domain ())('. 9t", . . . ). From this it appears that in 11) the radicals V /''■• \ '/£,,... do not admit of multiplying every term by an arbitrary root of unity, as indeed is already evident d priori since otherwise x„ would have not p, but p 9 values. A still further transformation of 12) is possible. We have f/R;= G e t/Rf = cS(i? ) • f/RJ. From § 221 there are alterations in the V u V at ... V a _ 1 which convert R into R k and consequently */ R into «>" "* ' *J~R k . The form of the exponent of oj evidently involves no limitation. At the same time the a? becomes THE ALGEBRAIC SOLUTION OF EQUATIONS. 263 and since R x becomes R k + l , it follows that %/ R 1 becomes If now we apply these transformations to the equations above, we obtain We can therefore also write x =G +tfR + Z/R h +\ / i^-+... + ^(Bo) • Z/R7 + &(-«*) • &I&+ MX**) ■ Z/I& + • • • + c^o) . #js? + ^(r*) • Vr7 [ + 0,(jr,») • \/iv 2 + . . . Theorem X. ITje roote o/ a solvable equation of prime degree p can be written in either of the two forms 12) or 13). In 13) v -ft/.- , V -^2*5 • • • are rational functions of \/ R . The values R , R k , R 2k , . . . R( p-i _Ak are roots of a simplest Abelian equation, the group of which is com- jyosed of the powers of 9=(R R k R 3k ...) Its roots are connected by tlie relations 14) VB k =f(B ) ■ ViC", / JR will become some w^v -fro > an <3 from 14) \/R k .z=f(R ) . V-Ko** becomes iiAic equation - . $ 226. We have already seen in Chapter IX, § 153 that every special, or affect, equation /(.f) = is completely characterized by a single relation between its coefficients or between its roots. Sup- pose that in any particular case the relation is More accurately speaking, it is not the function

-2 = 0, for example, requires very different means, according as s/'l is or is notincluded in the rational domain. The rational domain can be defined on the one hand by assigning the elements 91', l)i", . . . , from which it is constructed. Or we may construct the Galois resolvent equation and determine one of its irre- ducible factors in the rational domain. The latter does not, to be sure, entirely replace the assignment of 3t', "M", . . . , but it furnishes THE GROUP OF AN ALGEBRAIC EQUATION. 207 everthing which is of importance from the algebraic standpoint for the equation considered. The determination of the irreducible factor gives at once the group of the equation; in the n\ factors ^ — (".-, »i + I',-, *a + • • • + ",-„ a?») of which the Galois resolvent is composed, we have only to regard the u's as undetermined quantities, and to form the group of the tt's which permute the factors of the irreducible factor among them- selves. It must be always borne in mind that from the algebraic stand- point only those equations have a special character, according to Kronecker an affect, for which the Galois resolvent of the (n!) tb degree is factorable. § 227. On account of the intimate connection between an equa- tion and its group, we may carry over the expressions " transit ive," '•primitive" and "non-primitive," "simple" and "compound" from the group to the equation. Accordingly we shall designate equations as transitive, primitive or non- primitive, simple or compound, when their groups possess these several properties. Conversely, we apply the term "solvable," which is taken from the theory of equations, also to groups, and speak of solvable groups as those whose equations are solvable. Since, however, an infinite number of equations belong to a single group, this usage must be justified by a proof that the solution of all the equations belonging to a given group is furnished by that of a single one among them. This proof will be given presently (Theorem V). In the first place we attempt to reproduce the properties of the groups in the form of equivalent algebraical properties of their equa- tions. We have already (§ 156) Theorem II. If an equation is irreducible, its group is transitive; conversely, if the group of an equation is transitive, the equation is irreducible. § 228. To'determine under what form the non-primitivity of the group reappears as a property of the equation, we recur to the treat- ment of those irreducible equations one root of which was a rational function of another. The equation of degree m> reduced to v equa- 2C8 THEORY OF SUBSTITUTIONS. tions of degree m, the coefficients of which were rationally expressi- ble in terms of the roots of an equation of degree v (§ 174). We arrive in the present case at a similar result. Suppose that the group G of the equation f(x) = is non-prim- itive; then the roots of the equation can be distributed into v sys- tems of m roots each such that every substitution of the group which converts one root of any system into a root of another system converts the entire former system into the latter. We take now for a resolvent any arbitrary symmetric function of all the roots of the first system 1) V\ — '-5 \p?n ? *^i2» • • • •* "i"' ' and apply to S all the substitutions of G. Since G is non-primi tive, the entire system x u , x n . . . x lm is converted either into itself or into one of the other systems. There are therefore only v values of y V\ —'J (pCllI «^12 5 • • • •*']»i/> V/j — 0^3^215 ^*22 ? • • • *~tm)l ") \ Vz ~ &[X ZXi ***32J • • • X Zm)i Consequently y is a root of an equation of degree v 3) Ky)=o, the coefficients of which are unchanged by all the substitutions of G, and which are therefore, from Theorem I, rationally known. If c (,//) --0 lias been solved, i. e., if all its roots y lt y 2i . . . y v are known, then all the symmetric functions of every individual sys- t' in are also known. Tor each of these functions belongs to the same group as the corresponding y, and can therefore be rationally expressed in terms of the latter and of the coefficients of f(x). If we denote, in particular, the elementary symmetric functions of **-a\ > XgQ i • • • 3 am Oy Vi/a), — s ',„'//a), then the quantities a •„, . a a .., . . . x a „, are the roots of the equation 4) ar—S l (vJar- 1 +S a foJar-*—.. . ±S a (y a ) = THE GROUP OF AN ALGEBRAIC EQUATION. 209 Consequently f(x) can be obtained by eliminating y from 3) and 4), and we have V f(x)== JT'[x m S 1 {y a ) Jt-'"- 1 + S,(y a) .,---— . . . ± SJy a )~] = 0. a = 1 Conversely, if we start from the last expression, as the result of eliminating y from 3) and 4), then the group belonging to f(.r) = is non primitive, if we assume that 8) and 4) are irreducible. For we form first a symmetric function of the roots of 4). This is rational in y a . We denote it by F(y a ). Again we form the product 5) [u— F{y$\ \u—F(y$\ . . • [u— i%„)] for all the roots of 4). This product is rationally known; for its coefficients are symmetric in ?/, , y,. . . . y v , and are therefore ration- ally expressible in the coefficients of 3). Accordingly 5) remains unchanged by all the substitutions of the group, i. e., every substi- tution of the group interchanges the linear factors of this product only among themselves. If therefore F(y a ) can be expressed in terms of the ac's in only one way, it follows that the group converts the symmetric functions of .r a] ..r a . v am into those of another system. The group is therefore non- primitive. But if the roots of fix) are different from one another, the assumption in regard to F(y a ) can be realized by § 111. Theorem III. The group of an equation of degree m>, which is obtained by the elimination of y from the two irreducible equations 3) . The groups of all the factors F t (^), ... of F{=) therefore differ from one another only in the particular designation of their elements. Theorem IV. If a special equation f(x) = is charac- terized by the family of 9) = — factors 10) i^)=0, every one of which can serve as the Galois resolvent of the special equation. All the roots of 10) are rational functions of every one among them, and in terms of these all the roots of f(x) =0 can be rationally expressed. The transition from f(x) = to F l (:) = has its counterpart in the transition from G to the simply isomor- phic group Q (§ 129) of F^). 2 l 2 THEORY OF SUBSTITUTIONS. Since the construction of 10) depends only on the group G, and not on the particular nature of 9), this same resolvent belongs to all equations which are characterized by functions of the same family with 9). If one of these equations has been solved, then a?j, x 2 , . . . <„ and consequently I, are known. The equation 10) is therefore solved, and with it every other equation of this sort. We have then the proof of the theorem stated in § 226: Theorem V. Given an equation f(x) = 0, the coefficients of which belong to any arbitrary rational domain, the adjunction of either cr, = or

presents itself implicitly as a root of an equation which is regarded as solvable. For example, in the problem of the algebraic solution of equations the auxiliary equation is of the simple form y p — A(x 1 ,x 2 , . . . x n )=0. Here y is regarded as known, i. e., we extend the rational domain of f(x) — by adjoining to it every rational function of the roots of which any power belongs to the domain. The actual solution of the auxiliary equations does not enter into consideration. < It is a natural step, when an irreducible auxiliary equation is regarded as solvable, to adjoin not one of roots d>, but all of its roots to the domain of fix) = 0. These roots are the different val- ues which t'l',. .*-,, . . . .*■„) assumes within the rational domain. For to find the auxiliary equation which is satisfied by 4\ we apply to <•, all the /• substitutions of the group G and obtain, for example, m distinct values The symmetric functions of these values, and therefore the coeffi- cients of the equation THE GROUP OF AN ALGEBRAIC EQUATION. 273 1 2 ) g () = (

) = are known within the rational domain of f(x) = 0, and 12) is the required auxiliary equation, the solution of which is regarded as known. Now given the equation f(x) = 0, characterized by the group G, or by any function where the a's are undetermined constants. The question then arises, what the group of f(x) = becomes under the new conditions. The adjoined family of functions was originally that of • • • H m , of 0, , 3 , . . . the result is in every case the same series in a new order; for m — ^r-'+ . . . =0, the coefficients of which are rational in the rational domain of f{x), and the roots rational functions of x l , x 2 , . . . x„, then G reduces to tlie largest self- conjugate subgroup V of G ichich leaves

m all unchanged. If G is a simple group, T= 1. Only in case G is compound is it possible by the solution of an auxiliary equation to reduce the group to a subgroup different from identity, and conse- quently to divide the Galois resolvent equation into non-linear fac- tors. § 231. We consider these results for a moment. If the general equation of the n th degree /(a?)=0 is given, the corresponding group G is of order r = nl. This group is compound, the only actual self conjugate subgroup being the alternating group (§92). If we_take for a resolvent <\ = V - , where J denotes, as'usual, the discriminant of f(x), then the resolv- ent equation becomes 12') <>■— J = 0, and /' is the alternating group. After adjunction of the two roots of 12') the previously irreducible Galois resolvent equation divides into two Jconjugate factors of degree \n\, and only such substitu- tions can be applied to the resolvent £ = 0,05, + O^aJj-f- ... + «„ X H as leave \/ J unchanged and therefore belong to the alternating group. For n > 4 the alternating group is simple. If there is an m- THE GROUP OF AN ALGEBRAIC EQUATION. 275 valued resolvent 4> •> its values 4'u 4'n • • • 4'm are obtained by the solution of an equation of the m th degree. On the adjunction of these values, or of Z = j8, ft +&&+... + P m 4' m the group of the given equation reduces, by Theorem VI, to the identical substitution. The equation f(-r) = is therefore solved; for all functions are known which belong to the group 1 or to any other group. The investigations of Chapter VI show, however, that no reduction of the degree of the equation to be solved can be effected in this way, since if n > 4, the number m of the values of 4) is solved, as soon as any arbitrary resolvent equation of a degree higher than the second is solved. There are, however, no resolvent equations the degree of which is greater than 2 and less than n. Moreover, if w= 6, there is no resolvent equation of the n"' degree essentially different from f(x) — 0. For n = 6 there is a distinct resolvent equation of degree 6. One other result of our earlier investigations, as reinterpreted from the present point of view, may be added here: Theorem VIII. The general equation of the fifth degree has a resolvent equation of the sixth degree. § 232. We return now, from the incidental results of the pre- ceding Section, to Theorem VI, and examine the group of the equation 12) fir (^)==(^_^ 1 ) (4>—4> 2 ) . . . {4>—4> m ) = 0, thp roots (/•,, c'' : . . . . (/•„, of which were all adjoined to the equation /(a?) = 0. The order of the group of 12) is most easily found from the fact that it is equal to the degree of the irreducible equation of which w = /'l V\ + T% l + • •■ • + Ym of the group of every resolvent equation is the same as that of /(.»') = 0, so that no simplification can be effected in this way. We actually obtain the group of 12) by the consideration that it contains all and only those substitutions among the as's which do not alter the nature of f(x) = 0. If therefore we apply to th iU iU i'i r 1 J V2 J V 8 > • • • fm all the substitutions of G, the resulting permutations of the ^'-'s form the group required. All the r substitutions thus obtained are not however necessarily different; for all the substitutions of l 1 leave all the elements unchanged. From this, again, it follows that the order of the group K of 12) is v = r:r'. In the same way we recog- nize that K is (1 — r) fold-isomorphic to G. With the notation of § 86, K is the quotient of G and l'- K = G:I\ Theorem IX. // the group G of f(x) = is of order r y and contains a self -conjugate subgroup I' of order r', and if G reduces to /' on the adjunction of all the roots 11) of 12) 0(0) = O, then the group K of the latter equation is of order v =r: r'. K is the quotient of G and r and is (1 — r)-fold isomorphic to G. By a proper choice of the resolvent

= r: r' run be constructed, the roots of which are all rational functions of a single one among them, and which possesses the properly /hat the adjunction of one of its roots to fix) = reduces the group G to /'. §233. Theorem XI. If r is a maximal self- conjugate subgroup of G, then the group of h(x) = is a transit! re, simple group. Conversely, if f is not a most extensive self -conjugate sub- group of G, then the group of h(x) = is compound. We denote the group of h(x) = by G'. Its order is v = r: r'. We assume that G' contains a self- conjugate subgroup /'', of order r v From Theorem IX G' is r'-fold isomorphic to G. From the results of § 73 it follows that the subgroup J of G, which corresponds to the group T', is a self- conjugate subgroup of G and is of order •/ /■'. J is, then, like /', a self conjugate subgroup of G, and their orders are respectively v' r' and r'. We show that F is con- tained in J. This follows directly from the construction of G' (§ 232), in accordance with which the substitution 1 of G' corresponds to all the substitutions of G which leave the series 11) unaltered. /' in G therefore .corresponds to the one substitution 1 of G'. Accordingly if G' is compound, then P is not a maximal self-conjugate subgroup of G. The converse theorem is similarly proved from the properties of isomorphic groups. In these last investigations we have dealt throughout with the group of the equation, but never with the particular values of the coefficients. If therefore two equations of degree ;/ have the same group, the reductions of the Theorem X are entirely independent of the coefficients of the equations. The coefficients of //(/) will of course be different in the two cases, but the different equations h(y) = all have the same group, and every root of any one of these equations is a rational function of every one of its roots. This com- mon property relative to reduction, irhicic holds also for the further 278 THEORY OF SUBSTITUTIONS. investigations of the present Chapter, is the chief reason for the collection of all equations belonging to the same group into a family. £ 234. We observe further that with every reduction of the group there goes a decomposition of the Galois resolvent equation, while the equation f(x) = need not resolve into factors. Collecting the preceding results we have the following Theorem XII. If the group G of an equation f(x) — is compound, and if G, (?] , Gr 2 , . . . Gr„, 1 is a series of composition belonging to G, so that every one of the groups (?,, G 2 , • • • G v , 1 is a maximal self-conjugate subgroup of the preceding one, further if the order of the several groups of the series are r i r i 5 r 2l ' • • r vi t, then the problem of the solution of f(x) = can be reduced as fol- iates. We have to solve in order one equation of each of the de- grees r r\ r 2 rv_, t*! r 2 ?*3 r v the coefficients of which are rational in the rational domain deter- mined by the solution of the preceding equation. These equations are irreducible and simple, and of such a character that all the roots of any one of them are expressible rationally in terms of any root of the same equation. The orders of the groups of the equa- tions are respectively r r, rj r v _ l , , — .... , i v . r, r 2 r 3 r v The groups are the quotients G: (! x , G i :G 2 , G 2 : G 3 , . . . G„_ l :G„, G v :l. The equations being solved, the Galois resolvent equation, which teas originally irreducible and of degree r, breaks up successively into /■ /■ r r r,' tV r 8 ' "' 7/ T factors. After the last operation f (x) = is therefore completely solved THE GROUP OF AN ALGEBRAIC EQUATION. 279 § 235. The composition of the group G of an equation fix) = is therefore reflected in the resolution of the Galois resolvent equa- tion into factors. We turn our attention for a moment to the ques- tion, when a resolution of the equation f(x) = itself occurs. It is readily seen that, in passing from G a to G a+i in the series of com- position of G, a separation of f(x) into factors can only occur when G a +\ does not connect all the elements transitively which are con- nected transitively by G a . The resulting relations are determined by §71. G a is non-primitive in respect to the transitively con- nected elements which G a + 1 separates into intransitive systems. Starting from G, with an irreducible f(x) = 0, suppose now that G l , G-,, . . . G a are transitive, but that G a+l is intransitive, so that by § 71 G a is non- primitive. Then at this point f(x) separates into as many factors as there are systems of intransitivity in G a + li But (again from §71), all the elements occur in G a+l . We arrange, then, the substitutions of G a in a table based on the sys- tems of intransitivity of G a + 1 . Suppose that there are //. such sys- tems, so that f(x) divides into fi factors. Then we take for the first line of the table all and only those substitutions of G a , which do not convert the elements of the first system of intransitivity into those of another system. The substitutions of this line form a group, which is contained in G a as a subgroup. Its order is there- fore kr a+1 . The second line of the table consists of all the substi- tutions of Ga which convert the first system of intransitivity into the second. The number of these is also kr a+i . There are //. such lines, and they include all the substitutions of G a . Consequently K 'a + 1 i. e., the number ,u of the factors into which f(x) divides is a divi- r sor of tlte number of the factors into ivhich the Galois resolv- T a+ 1 ent equation divides at the same time. A similar result obviously occurs in every later decomposition. The decomposition can therefore only take place according to the scheme of Theorem III. The several irreducible factors are all of the same order. § 236. Thus far we have adjoined to the given equation /(^)=0 2N0 THEORY OF SUBSTITUTIONS. the root i) (

(z 2 ), . . ■ — — ^2)] ■ . ■ [? — c'(c), and the n values ^(z^, (z 2 ), ■ • • ^''(z^) coincide in sets of q each. THE GROUP OF AN ALGEBRAIC EQUATION. 281 "With, a slight change in the notation for the 2's we can therefore \v cite Z) *=#«•) = f«') =... = ^(0, ( * m -'" Since which is isomorphic to 1\ To the substitution 1 in T correspond in F the substitutions of the subgroup ^ of order d { which only interchange z\,z' 2 , . . . z' q among themselves, z'\, z" 2 , . . . z" q among themselves, and so on. F and T are (1 - — d^-fold isomorphic. If we coordinate all the substitutions of G and T which leave cr, unchanged, and again one substitution each from G and T which converts or to an inclu- ded family. For every such function is rationally knowu, as soon as the z t , z 2 , . . . z^ are adjoined to the equation f(x) = 0. Conversely, if we adjoin to the equation g (z) = all the roots •of f(x) = 0, it follows by the same reasoning that there is a function 15) • w (z u z 2 , ...Zp) =p (x 1 , a^,...a?„), 282 THEORY OF SUBSTITUTIONS. such that every function of z x ,z.±, . . . z^ which can be rationally expressed in terms of .»', , x.,, . . . .<•„ belongs to the family of u> or to an i lie haled family. Since now /< is rational in the z's, it follows from the above prop- erty that 16) Po = R(p), where R is a rational function; and since at is rational in the aj's, it follows that 17) io — R v (, so that /' reduces to J. But the proof just given was necessary to exclude the possibility of any further reduction. r v. If we write — = -j-—- 1 , it follows that if the second adjunction reduces the order r of G to its > Ul part, then the first adjunction also reduces the order r t of r to its > th part. Theorem XIII. The effect of the adjunction of all the roots of any arbitrary equation 13) on the reduction of the group Goff(x)=0 can be equally well produced by the adjunction of all the roots of an equation 12) which is satisfied by rational func- tions Of SCj, x 2 , . . . x n . In spite of removal of apparent limitations, we have therefore not departed from the earlier conditions, where only the adjunction of rational functions of the roots was admitted. §237. Theorem XIV. // f(x) = 0, ?(«) = () are two equations, the coefficients of which belong to the same rational domain, and which are of such a nature that the solution THE GROUP OF AN ALGEBRAIC EQUATION. 283 of the second and the adjunction of all its roots to the first reduces the th part. The group of f(x) = 0, like that of g(z) = Q, is compound, and v is a factor of composition. Those rational functions of the roots of one of the two equations, by which the same reduction of its group is accomplished as by the solution of the other equation are rational in the roots of the latter. As we see, the group of fix) = can be reduced by the solution of an equation g(z)=0, although the roots of the latter are not rational functions of x x ,x 2 ,... x„ . It is only necessary that thore should be rational functions of z l , z 2 , . . . z^ which are also rational functions of x, , x. 2 , . . . x n . From the preceding Theorem follow at once the Corollaries Corollary I. If the group G of the equation f(pc) = is simple, the equation can only be solved by tlie aid of equations with groups the orders of which are multiples of the order of G. For since G reduces to 1, the v of Theorem XIV must be taken equal to the order of G. Corollary II. If the group G of f(x) = can be reduced by the solution of a simple equation g(z) = 0, then z n z 2 , . . . z^ are rational functions of the roots of f(x) = 0. For in this case v is equal to the order of the group of g (z) — 0. After the reduction this is equal 1. Consequently '/'", —a x z^ + a.,z, -f- . . . + a^Zfj, = ,■■■ 2>), in ivhich the roots of the two equations are separated. For, if we denote the corresponding Galois resolvents by I and X, and the irreducible resolvent equations of f(x) = and g(z) = by tf(*)=0, G(:) = 0, the degrees r and r' of F and G are equal to the orders of the respective groups. Now ' ] . ■'•., '•„ can be rationally expressed in terms of ?, and Zi,Z 2 , . . . Zy, in terms of ~, so that can be so reduced by the aid of F = and G = that its degree becomes less than r in - r and less than r' in %. Then the two equations *(* :) = 0, /•'(f) = have a common root. Consequently, if we add X to the rational domain, the resolvent F(:) becomes reducible, since otherwise the irreducible equation of the r th degree would have a root in common THE GROUP OF AN ALGEBRAIC EQUATION. 285 with an equation of a degree less than r. The only exception occurs when #(£, %) is identically 0. If this does not happen, the adjunction of all the roots of g (z)= or that of ~ breaks up the resolvent of /(a?) = into factors, and we have therefore the case of the last Section. We can effect the same reduction by the adjunction of a rational function / of ■> \- x 2 , . . . and we have X{x u x 2 , r„) = i!>(z x , z 2 , . . . zj. If several such relations exist, they can all be deduced from one and the same equation. The latter can be easily found, if we select a function y such that all the others belong to an included family. On the other hand if *(,-, 1') is identically 0, it follows that the coefficients in the polynomial '!'(:. X) arranged according to powers, of f vanish, so that we have equations of the form yj') z ,(:,. :_,, . . . ^) = 0, and similarly, if &(%, Z) is arranged in powers of '~, v y ,r i > •* 2 ? • • ■ «*v) ~~ '-'• These equations can actually make . . . ALGEBRAICALLY SOLVABLE EQUATIONS. 287 This condition is also sufficient, as has already been shown in §§ 110, 111, Theorems X and XII. Not that every function belong- ing to (7 A on being raised to the (px) th power gives a function belonging to G^—i] but some function can always be found which has this property, as soon as the condition above is fulfilled. We have then Theorem I. In order that the algebraic equation f(x) — may be algebraically solvable, it is necessary and sufficient that the factors of composition of its group should all be prime numbers. § 240. By the aid of Theorem XII, § 110 we can give this theo- rem another form Theorem II. In order that the algebraic equation f{x) = may be algebraically solvable, it is necessary and sufficient that its group should consist of a series of substitutions ^1 M 1 *2> '3> • • • 'vi 'v + 1 which p>ossess the two following properties: 1) the substitutions of the group G\ = {1, £ n t 2 , . . . h\ arc commutative, except those which belong to the group 6rx_j = \ 1, f M t.,, . . . t\_ 2 , t\-i\, and 2) the loivest poiver of f A , which occurs in G\_ x has for its exponent a prime number (cf. also §91, Theorem XXIV). § 241. Again the investigations of § 94 enable us to state The- orem I in still a' third form. It was there shown that if the prin- cipal series of G 2) G, H , J, K, ... 1 does Lot coincide with the series of composition, then 1 ) can be obtained from 2) by inserting new groups in the latter, for example between H and J the groups H', H", . . . HM. Then the factors of composition which correspond to the transitions from H to H', from H' to H", . . . from H A) to J are all equal. Accordingly, if all the factors of composition belonging to 1 ) are are not equal, then G has a principal series of composition 2). We saw further (§95) that, if, in passing successively from H, H', H", ... to the following group, the corresponding factors of composition were all prime numbers, (which then, as we have just 288 THEORY OF SUBSTITUTIONS. seen, are all equal to each other), and only in this case, the substitu- tions of H are com mutative, except those which belong to J. From this follows Theorem III. In order that the algebraic equation /(aj) = may be algebraically solvable, it is necessary and sufficient that its principal scries of composition G, H, J, A, . . . 1 should possess the property that the substitutions of every group are commutative, except those which belong to the next following group. The substitutions of the last group of the series, that which pre- cedes the identical group, are therefore all commutative. § 242. Before proceeding further with the theory, we give a few applications of the results thus far obtained. Theorem IV. If a group V is simply isomorphic with a solvable group G, then F is also a solvable group. From § 96 the factors of composition of G- coincide with those of /'. Consequently Theorem IV follows at once from Theorem I. Theorem V. If the group /' is multiply isomorphic with the solvable group G, and if to the substitution 1 of G corresponds the subgroup - of I\ finally if - is a solvable group, then /' is also solvable. The factors of composition of /' consist, from § 96 ? of those of G and those of -. Reference to Theorem I shows at once the validity of the present theorem. Theorem VI. If a group G is solvable, all its subgroups are also solvable. We write as, usual -~i = "i #i + a i&2 +•••+«« ■''» j apply to £] all the substitutions of G, obtain r,, .%, . . . ; r , and form g(*)=(e_ *,)(*— eo ■..(*— *.)• It is characteristic for the solvability of G that g {:) can be resolved into linear factors by the extraction of roots. If now 11 of order r is a subgroup of G, and if the applica- ALGEBRAICALLY SOLVABLE EQUATIONS. 289 tion of H to I, gives rise to the values r, , _-,, . . . * then these are all contained among ?,, --_., . . . :,.. Consequently fe(|) = (e-^)(^-^)...(?-f n ) is a divisor of {?(£). Then h{z) is also resolvable algebraically into linear factors, i. e., H is a solvable group. We might also have proved this by showing that all the factors of composition of H occur among those of G. Theorem VII. If the order of a group G is a power of a prime number p, the groiqi is solvable. The group G is of the same type as a subgroup of the group which has the same degree n as G and for its order the highest power p f which is contained inn! (of. §§ 39 and 49). That the latter group is solvable follows from its construction (§ 39), all of its fac- tors of composition being equal to the prime number p. It follows then from Theorem VI that G is also solvable. Theorem VIII. If the group G is of order r=pfpfp z yp t 8 . . . where Pi,p 3 ,p 3 , p t , . . . ore different prime numbers such that Pi > PfPz y Pi & ■'•■■> Pi> Pz y P* ■•-■> Pi> Pi ■■•■> then G is solvable.* We make usex>f the theorem of § 128, and write r = p*q, where then p t > q. G contains at least one subgroup H of the order'p,". If we denote by kp { + 1 the total number of subgroups of order p, a contained in G, and by p x H the order of the maximal subgroup of G which is commutative with H, then r = p*i(kp l -\- 1). Since r — p x a q and q < p, , we must take k — and r = p { a i. That is, G is itself commutative with H. By the solution of an auxiliary equation of degree q, with a group of order q, we arrive therefore at a function belonging to the family of H, and the group G reduces to i/(£ 232), Theorem X). From Theorem VII the latter group is solvable. Accordingly, if the auxiliary equation is solvable, the group G is solvable also. The group of the auxiliary equation with the order q =p/p 3 y .. . admits of the same treatment as G. Its solvability therefore follows *L. Sylow: Math. Aim. V, p. 585. 19 290 THEORY OF SUBSTITUTIONS. from that of a new auxiliary equation with a group of order p 3 Y p/ . . . , and so on. § 243. We return to the general investigations of § 241. The transition from G to G l decomposes the Galois resolvent r equation into-=p, factors. The transition from G x to G-, decom- V poses each of these previously irreducible factors inte — =J9 2 new factors, and so on. Since f(x) = was originally irreducible, but is finally resolved into linear factors, it follows from § 235 that once or oftener a reso- lution of /(.*■) or of its already rationally known factors will occur simultaneously with the resolution of the Galois resolvent equation or of its already known rational factors. The number of factors into which f(x) = resolves, which is of course greater than 1, must from § 235, be a divisor of the number of factors into which the Galois resolvent equation divides. In the case of solvable equa- tions the latter is always a prime number Pi,p 2 ,Pa, ■ • ■ Conse- quently the same is true of f(x) = 0. All prime factors of the degree n of the solvable equation f(x) = are factors of composi- tion of the group G, and in fact each factor occurs in the series of composition as often as it occurs in n. To avoid a natural error, it must be noted that if in passing from G to G K the polynomial f(.r) resolves into rational factors one of which is f\(x), this factor does not necessarily belong to the group G K - It may belong to a family included in that of G\. The number of values of f\(x) is therefore not necessarily equal to r:i\. It may be a multiple of this quotient. And the product f'\(x) f"\(x) ... of all the values of f\(x) is not necessarily equal to /(.'). but may be a power of this polynomial. We will now assume that n is not a power of a prime number p, so that n includes among its factors different prime numbers. Then different prime numbers also occur among the factors of composi- tion of the series for G, and consequently (§ 94, Corollary I) G has a principal series G, H,J,K,... M, 1. Suppose that in one of the series of composition belonging to other groups ALGEBRAICALLY SOLVABLE EQUATION8. -J'.* 1 3) H', H", . . . H^> occur between H and J. Since n includes among its factors at least two different prime numbers, f(x) must resolve into factors at least twice in the passage from a group of the series of composition to the following one. Since the number of the factors of f(x) is the same as the factor of composition, and since the latter is the same for all the intermediate groups 3), the two reductions of fix) cannot both take place in the same transition from a group // of the prin- cipal series to the next following group J. It is to be particularly noticed, that all the resolutions of f(x) cannot occur in the transition from the last group M to 1, that is, within the groups 31', M", ...M^~ l \l, following M in the series of composition. At least one of the resolu- tions must have happened before M. Suppose, for example, that the first resolution occurs between H' and H". Then it follows from § 235 that H' is non-primitive in those elements which it connects transitively, and that H" is intransitive, the systems of intransitivity coinciding with the system of non- transitivity of H'. The same in- transitivity then occurs in all the following groups H'", . . . H^\ and likewise in the next group J of the principal series, which by assump- tion is different from 1. Suppose that J distributes tbe roots in the intransitive systems x\ ,x' 2 ,... x'r, x'\ , x" 2 . . . x"r, . . . x^\ x ( p, . . . #«, these systems being taken as small -as possible. Then the expression f\(x) — {x — x\) (x — x f 2 ) . . . x — x',) becomes a rationally known factor of f(x), which does not contain any smaller rationally known factor. Since from the properties of the groups of the principal series G~ l JG = J, all the values of f\(x) belong to the same group J. They are there- fore all rationally known with f\(x). Of the values of f',\(x) we know already fx(x) = (X — X\) (x — x'. 2 ) . . . (x — .r' ). f\(x) = (x—x'\) (x—x",) . . . {x — x",), f K W(x) = (x—x^) (x—x 2 C">) . . . {x—xjrt). 292 THEORY OF SUBSTITUTIONS. If there were other values, these must have roots in common with some / A la ' (.r). Then J\ {a '(x) and consequently /" A (.c) would resolve into rational factors. This being contrary to assumption, f\(x) has only m = - values, and is therefore a root of an equation of degree m. If this equation is n ?(y) (/y-A) (y— /"a) • • • (y-f\" n) ) = 0, then f(jr) is the result of elimination between 4) and 5) f\(x) = x<— Uy'y- 1J r Uv')^- 2 — ■ ■ • = 0, where M^) = x' 1 + x' 2 +x' 3 -\- ...+a-' ; . 4 ■.. (//') = x\ x', + X\X% + . . . + X'i _i X' { , so that 0,, c\,, . . . are rationally expressible in terms of f\. Since /(#) is the eliminant of 4) and 5), it follows from § 228 that the group of f(x) — is non-primitive. These conclusions rest wholly on the circumstance that J belongs to the principal series of G, and that accordingly G~^J G = J. It is only under this condition that all the values of f\(x) which occur in the rational domain of f(x) = are rationally known. This shows itself very strikingly in an example to be presently considered. Theorem IX. If the degree n of an irreducible algebraic equation is divisible by two different prime numbers, then n can always be divided into tiro factors n = im, such that the given equa- tion fix) = resolves into m new ones j\(x) = 0,f\(x) = 0,... fW(x)=0, which are all of degree i, and the coefficients of which are obtain- able from known quantities by the solution of an equation of degree m.* The group of the equation /(as) = is non-primitive. For the purpose of comparison we consider the solution of the general equation of the fourth degree, to which, since 4 = 2 a , the preceding results are not applicable. It appears at once that both of the resolutions of the polynomial into linear factors take place in the domain belonging to the last group of the principal series M. M', M", ... 1. The series of the equation consists of the follow- ing groups: ♦Abel: Oeuvres completes II, p. 191. ALGEBRAICALLY SOLVABLE EQUATIONS. 293 1) the symmetric group; 2) the alternating group: 3) [1, {x x x 2 ) i. <■;.<, i. !••■ .-'■ I ''■'■,) (•'•,■'•,) (■'■.■'•> |; 4) [1, (a-, .*■..) (as ,.p 4 )], 4: , )[l 1 {x 1 x i )(x 2 x i )'\, or 4") [1, (.r,.rj {,■.,-, ] : 5) the group 1. The principal series consists of the groups 1 ), 2 ), 3), 5). The passage from 3) to 4) and that from 4) to 5) both give the prime factor 2. The group 4) is the first intransitive one. For this/Cr) resolves into the two factors (x — a^) (x — x 2 ) and (x — sr 8 ) (•»• — -''J- But since the group 4) does not belong to the principal series, all the six val uesof (as — .<,) (x — a? 2 ) are not known. If we had chosen the group 4') instead of 4), we should have had the two factors (x — a:,) (x — x 3 ) and (x — x 2 ) (x — a? 4 ), and so on. The product of these six values give the third power of /(a?)=(as — x x ) (x — x 2 ) (x — x 3 )(x — a%). We can therefore, to be sure, resolve f(x) into a product of two factors of the second degree. But the coefficients of every such 4 factor are the roots not of an equation of degree -~ = 2, but of an equation of degree 6. If we consider further the irreducible solvable equations of the sixth degree, it appears that these are of one of two types, accord- ing as we eliminate y from x 2 — f (y)x + f, (y) = 0, y 3 — c, y' 1 + c 2 y—c 3 = 0, or from x 3 —f 1 (y)x' 2 -\~My)x—f 3 (y) = 0, y' — c, y + e 2 = 0. § 244. The preceding results enable us to limit our considera- tion to those equations f(x) = the degree of which is a power of a prime number p. For otherwise the problem can be simplified by regarding the equation as the result of an elimination. Further- more we may assume that such a resolution into iactors as was con- sidered in the preceding Section does not occur in the case of our present equations of degree /> A , since otherwise the same simplifica- tion would be possible. We assume therefore that the group of the equation is primitive, thus excluding both the above possibilities. With this assumption we proceed to the investigation of the group. Suppose that the degree of the equation is p K and that its principal series of composition is 2) G, H,J,K,... M, 1, 294 THEORY OF SUBSTITUTIONS. In passing from G through H, J, . . . to M, no resolution of f(x) into factors can occur. Otherwise we should have the case of th° last Section, and G would be non- primitive. The passage from G to .1/ "prepares" the equation f(x) for resolution, but does not as yet resolve f(x) into factors. The /. resolutions of the equation of degree p A therefore occurs in passing from the last group of the principal series to identity, that is, in M, M', M", . . .M K ~\ 1. Accordingly we must have *>,/. The application of §94, Corol- lary IY shows that all the substitutions of M are commutative. The equation characterized by the family of M is therefore an Abelian equation of degree p K (§ 182). From § 94 there belongs to every transition from one group to the next in the last series the factor of composition p, so that the order of M is equal to p K . Again M can be obtained by combining ?. groups which have only the identical operation in common, which are similar to each other, and are of order p. Suppose that these are From the above properties it appears that every one of these groups is composed of the powers of a substitution of order p S, Si, S 2 , . . . S K _ ] , and that on account of the commutativity of the groups (cf. § 95) we must also have sS sp" = af s a " («, /9 = 0, 1, ... y. — 1). Consequently every substitution of M can be expressed by .S' Sj s 2 . . . s K _ j , and from the same commutative property Every substitution of the group M is of order p. Our Abelian equation therefore belongs to the category treated in § 18G, and its substitutions are there given in the analytic form / \z n z.., . . .z K «, + «,, z, + «,, . . .z K + a K \ (mod. p). The symmetric occurrence of all the indices z x , z 3 , . . . z K already ALGEBKAICALLY SOLVABLE EQUATIONS. '20," shows that in the reduction of M to 1 exactly /. resolutions of the polynomial f(x) will occur, as is also recognized if we write for example M'=\Zi,Z 3 ,z a ,...z K Z 1 ,Z 2 -\-a 2 ,Z a -\-a a ,...z K + a K (mod. /o. Af" = |z,,z 2 ,z 8 , ...z K z 1 ,z,,z. i + »-,,.. .z K -\- a K (mod.p), Accordingly /. = /, and we have as a first result Theorem X. The last group of the principal scries of a primitive, solvable equation of degree p K consists of the j> K arith- metic substitutions t=\z } , Z 2 , . . .Z K Z l -\-a l ,Z 2 -\-a 2 ,...Z K -\-a K \ (mod. p), the roots of the equation being denoted by X *i t *2...-* K ,(«x = 0,l,2, . . .p— 1). Since G, the group of the equation, is commutative with M, it follows from § 144 that G is a combination of arithmetic and geo- metric substitutions. We have therefore as a further result Theorem "XI. The group G of every solvable primitive equa- tion of degree p K consists of the group of the arithmetic substitu- tions of the degree p K , combined with geometric substitutions of the same degree u= z,,z.,,...z K a 1 z 1 + b x z, + . . . + c l z K ,a 2 z l -\-b 2 z 2 + . . . + c, z K , . . . (mod. p). § 245. Before proceeding further with the general investigation, we consider particularly the cases x = 1 and x = 2, the former of which we have already treated above. We consider first the solvable, primitive equations of prime de- gree p. We may omit the term " primitive,'' since non primitivit y is impossible with a prime number of elements. The group of the most general solvable equation of degree p must then coincide with or be contained in Cr= | z az-\-a.\ (a =1,2, ...p — 1; « = 0,1, .,. .p -1> (mod./-). We prove that the former is the case, by constructing the groups of composition from G to M and showing that all the factors of com- 2% THEORY OF SUBSTITUTIONS. position which occur are prime numbers. We divide p — 1 into its prime factors: p — 1 = g, q., . . . , and construct the subgroup p-1 2J= | z a.g, z + a x \ (a, = 1, 2, then the subgroup -j- a follows necessarily a 1, a I), and the substitution becomes iden- tical : 1 = J Z z\. If a substitution of G leaves one root X\ unchanged and if it con verts x K + l into x^, then every x v becomes x [lx K){v a» + a- For from / a / + " , ," a(/- + 1) + « , follows a .--,». — / , a '/.(). — p. - - 1), and the substitution is of the form z ( :> — /) z -+- /(/.- — //. -f- 1 1 . If a substitution of G leaves no root unchanged, and if it con- verts x K into .r^, then every .<-,. is converted into .»■,, ^ M „ A - For only in this case is there no solution / of the congruence \ cU + a, when a~ 1. If '• + 1 is to become />., then we must have />. = X-\- a. This gives <>. = ;i. — /, and the substitution is ! z z -\- fi — A | . These are precisely the same results which the earlier algebraic method furnished us. ALGEBRAICALLY SOLVABLE EQUATIONS. 297 Theorem XII. The general solvable equations of prime degree p are those of § 196. Their group is of order p(p — 1) and consists of the substitutions of the form s=\z az-\-a\ (a = l, 2, ...p — 1; a = 0, 1, . . .p — 1) (mod. p). Its factors of composition are all prime divisors of p — 1, each fac- tor ocurring as many times as it occurs in p — 1, and beside these p itself. § 246. We pass to the general solvable primitive equations of degree p 2 . As a starting point we have the arithmetic substitutions tf = | z x , z 2 z } + «i , z 2 + « 2 1 (mod. p), which form the last group M of the corresponding principal series. To arrive at the next preceding group, we must determine a substi- tution s which has the following properties. Its form is s= | z x , z 2 a x z x -f- b Y z 2 , a 2 z x -\-b 2 z 2 | (mod. p), and the lowest power of .s which occurs in M, and is therefore of the form t, must have a prime number as exponent. Since now all the powers of s are of the same form as s itself, the required power must be j z x , z 2 z x , Z 2 1 = 1. That is, the order of the substitution s must be a prime number. From these and other similar considerations we arrive at the fol- lowing results, * the further demonstration of which we do not enter upon. Theorem XIII. The general solvable, primitive equations of degree p 2 are of three different types. # The first type is characterized by a group of order 2p 2 (p — l) 2 , the substitutions of which are generated by the following: \z u Z % «! + «!, « a + «a| («i, «2 =0, 1, 2, . . .p — 1), i / i o •-> i\ (mod. p), |z,,z 2 a x z x ,a 2 z 2 \ (a,, a 2 , = l, Z, 6, . . ,p — 1), The groups belonging to the second type are of order 2 p 2 (p 2 — 1), and their substitutions are generated by the following : *C. Jordan : Liouville, Jour, de Math. (2) XIII, pp. 111-135, 20 '2 ( ,tS THEORY OF SUBSTITUTIONS. \z^z, z, + «, ,z 2 + a 2 1 (a,, 02=0,1, 2, ...p— 1), \z x ,z, azi+bez^bz. + az.^ (a,6=0, 1, . . .p — 1; 6u< not a,b=EO), |4,* «i f — A|, (mod.p). where e is any quadratic remainder (mod. p). The groups of the third type are of order 24 p"(p — 1). The form of their substitutions is different, according as p=l or p = 3 (mod, 4). In the former case the group contains beside the two substitutions |«n*i 3i + a n 3a + Os| (a,,a 2 = 0, 1, 2, . . .p — 1), / mod \ 2, ,2 2 azj,az 2 i (a=l, 2, 3...p — 1), aZso Me following four: \z x ,z 2 iz x ,—iz 2 \, \z x ,z 2 iz 2 ,iz 1 \, \z lf z 2 z 1 —iz 2 ,z l + iz 2 \, \z,,z 2 z ] + z 2 ,z l — z 2 \, where i is a root of the congruence i 2 = — 1 (mod. p). If p = 3 (mod. 4), the group contains the first two substitutions above, together with the following four: \z x ,Z 2 Z 2 , 3,|,. \Z 1 ,Z 2 SZ x ~\-tz.,,tz x — sz 2 \, \z x ,Z 2 — (l+st)z 1 + (s — Z 2, (t + s')Zi + (*t — 8 — t)z 2 \, \z t ,Z 2 SZ, + (1 +t)z.,, (t l)z, SjS 3 |, where s and t satisfy the congruence s 2 -\-t 2 = — 1 (mod. p). For p = 3 the first and second types, and for p = 5 the second type are not general. These types are then included as special cases in the third type, which is always general. § 247. We return from the preceding special cases to the more general theory. The same method which we have employed above in the case of p 2 can be applied in general to determine the substitutions of the group L which precedes M in the principal series of composition. L is obtained by adding to the substitutions t=\z x ,Z 2 , . . .Z K Z, + a x ,z 2 -\-a 2 , . . .z K + a K ) (mod. p) of M a further substitution 8=\z u z %i . . . z K a x Zr\-b l z 9 + * . .+c 1 z K ,a a z l +b&+ . . .+c a z K , . .. | (mod. p), ALGEBRAICALLY SOLVABLE EQUATIONS. 290 where the first power of s to occur among the f s has a prime expo nent. Since all the powers of s are of the same form as s itself, any power of s which occurs among the fs must be equal to 1. Consequently 8 must be of prime order. It is further necessary the group L = \ t, s\ should not become non-primitive. § 248. From the form to which the substitutions of G are re- stricted, we have at once Theorem XIV. All the substitutions, except identity, which belong to the group M= \z x ,Z 2 ,...Z K Z x + «, , z 2 + «,,... Z K -f a K affect all the elements. The converse proposition, which was true for z = 1, does not hold in the general case. For the element x Zl > Zn t , . , K is unaffected by 8— \z x ,z 2 ,...z a l z x -\-b x z l +...-\-c l z K +a x , a i z 1 J r b 2 z 2 -\-...-\- c z z, + «,,... | only in case the z congruences («i — l)^i + 612:.,+ ...+ c l z K + a 1 ^0, a 2 z 2 + (b 2 — l)z 2 + . . . + c 2 z K + a 2 — 0, b) (mod. p) a K z x + b K z,-\- ... +(c K — 1)^ + ^=0 are satisfied. Consequently, as soon as the determinant a x — 1 b x . . . c x a 2 b 2 — 1 . . . c 2 D = ;0 (mod. p), a K b K . . . c K — 1 the «, , a 2 , . . . a K can be so chosen that the congruences S) are not satisfied by any system z x ,z 2 , . . . z K . We consider now all the substitutions of the group G which leave one element unchanged. Since the distinction between the elements is merely a matter of notation, we may regard ;ro,o,...o as the fixed element. Then the substitutions which leave this element unchanged are r=\z u z i ,.'..z K a x z x J r b x z 2 J r ... -\-c x z K , a 2 z x -\-b,_z 2 -\- ...-{■ c 2 z K ...\. If we adjoin a' 0)0 ..„ to the equation, the group G reduces to /'. 300 THEORY OF SUBSTITUTIONS. Since all the substitutions of G are obtained by appending to those of /' the constants o n « 2 , . . . and since the «'s can be chosen in p K ways, it follows that the adjunction of a single root reduces G to its (p K ) th part. § 249. We will now consider the possibility that a substitution of G leaves * + l elements j\, % -,,... ;)t unchanged. Then the con- gruences S) of the preceding Section are satisfied by * + 1 systems of values z t , z a , . . . Z K Zi = ^ w , z 2 = :,'*>, ...z K = c««*) f (;. = 0, 1, 2, . . . x). We will however regard not the coefficients a, b, . . . c ; a of the substitution but the values £,(*>, r„ a) . . . f^W as known, and attempt to determine the substitution from these data. If now the determ- inant E is not =0 (mod. p), then the * systems T,), TV), . . . T K ) each of x -\- 1 congruences with the unknown quantities a, b, . . . c ; a T t ) (a, - 1) :,w + &, : 8 w + . . . + Cl :«« + «, =o, **«) a,:^' +(6 2 _l)C a W + ...-f-c 2 C (t W + a 9 =0, j r«) a, c,w + &. :.w + . . . + (c K — i) : K w + « K = o, have only one solution each, viz: L x ) «i = 1, &i = 0, t . . Cx = 0; «! = 0, L 2 ) Oj = 0, 6 2 = 1, . . . c 2 = 0; a 2 = 0, '1 '2 ■s 1 -» 2 • • • •= K c,w :,« . £«) a K = 0, 6 K = 0, . . .c K =l; « K = 0, and these solutions furnish together the identical substitution 1. We designate noiv a system of x + 1 roots of an equation for which E = (mod. p) as a system of conjugate roots. We have then Theorem XV. If a substitution of a primitive solvable group of degree p K leaves unchanged x-\-l roots which do not form a conjugate system, the substitution reduces to identity. ALGEBRAICALLY SOLVABLE* EQUATIONS. 301 ? 1 ? 2 • ' 1 ' 2 • ■*tt • • - K c oo f> w _ . . e«« If therefore we^ adjoin x -+- 1 such roots to the equation, the group G reduces to those substitutions which leave x + 1 roots unchanged, t. e., to the identical substitution. The equation is then solved. Theorem XVI. All the roots of a solvable primitive equa- tion of degree p K can be rationally expressed in terms of any x -J- 1 among them, provided these do not form a conjugate system. If we choose the notation so that one of the x -+- 1 roots is #o o ...o, the determinant becomes ±E = If the roots are not to form a conjugate system, then i£=0 (mod. p). The number r of systems of roots which satisfy this con- dition is determined in § 146. We found r = (p«_l) (p«— p)(p*— p 2 ) . . .{p*— p*- 1 ). Theorem XVII. For every root a?,,,^,...^ tve can deter- mine (ff« — 1) (p K —p) . • ■ (p K — P K ~') 1, 2, ... x systems of /. roots each such that these * + 1 roots do not form a con- jugate system, so that all the other roots can be rationally expressed in terms of them. The system composed of the x + 1 roots 3*0,0,0, ...J •^l.O.O.-.OJ 3*0,1,0... 0» ... 3*0,0,0, ... 1 is appropriate for the expression of all the roots. These results throw a new light on our earlier investigations in regard to triad equations, in particular on the solution of the Hes- sian equation of the ninth degree (cf. §§ 203-6 ). It is plain that we can construct in the same way quadruple equations of degree p 3 , and so on. THE LIBRARY UNIVERSITY OF CALIFORNIA Santa Barbara THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW. 9 25 OCT 9 9 5 50m-9,'6ti( G6338s8)9-182 3 1205 00084 9743 ^jlJZ UC SOUTHERN REGIONAL LIBRARY FACILITY AA 000 191 980 2