9RAW MA7H/8TA* 1 I9RARV -.,..,. < 1 THE UNIVERSITY OF CHICAGO SCIENCE SERIES Editorial Committee ELIAKIM HASTINGS MOORE, Chairman JOHN MERLE COULTER ROBERT ANDREWS MILLIKAN The University of Chicago Science Series, established by the Trustees of the University, owes its origin to a feeling that there should be a medium of publication occupying a position between the technical journals with their short articles and the elaborate treatises which attempt to cover several or all aspects of a wide field. The volumes of the series will differ from the discussions generally appearing in technical journals in that they will present the complete results of an experi ment or series of investigations which previ ously have appeared only in scattered articles, if published at all. On the other hand, they will differ from detailed treatises by confining themselves to specific problems of current interest, and in presenting the subject in as summary a manner and with as little technical detail as is consistent with sound method. FINITE COLLINEATION GROUPS THE UNIVERSITY OF CHICAGO PRESS CHICAGO. ILLINOIS Bgenta THE BAKER & TAYLOR COMPANY NEW YORK THE CUNNINGHAM, CURTISS & WELCH COMPANY LOS ANGELES THE CAMBRIDGE UNIVERSITY PRESS LONDON AND EDINBURGH THE MARUZEN-KABUSHIKI-KAISHA THE MISSION BOOK COMPANY SHANGHAI FINITE COLLINEATION GROUPS ih.j{. WITH AN INTRODUCTION TO THE THEORY OF GROUPS OF OPERATORS AND SUBSTITUTION GROUPS By H. F. BLICHFELDT Professor of Mathematics in Leland Stanford Junior University THE UNIVERSITY OF CHICAGO PRESS CHICAGO, ILLINOIS ^ COPYRIGHT 1917 BY THE UNIVERSITY OF CHICAGO All Rights Reserved Published April 1917 Composed and Printed By The University of Chicago Press Chicago. Illinois. U.S.A. PREFACE The theory of finite collineation groups (or linear groups) as developed at present is to be found mainly in scattered articles in mathematical journals, in addition to a few texts on group theory. The author has en deavored in the present volume to give an outline of the different principles contained in these publications, and has at the same time made an effort to depend upon a minimum of abstract group theoiy. In this and many other respects the present volume differs from Part II of the last book cited in 24 ; in particular, the present volume contains more of the theory of linear groups, though the student is referred to that Part II for lists of the invariants of the binary and ternary groups. No previous knowledge of the technique of group theory is required for the reading of the opening chapter, which develops the fundamental properties of linear transformations and linear groups. For the greater convenience of the student, an introduction to the theory of groups of operators and substitution groups is given in the second chapter; moreover, certain theorems and definitions from the more advanced parts of algebra needed throughout the book are stated in explicit form in an Appendix. The groups in two variables are determined in chap, iii by a modified form of a process due to Klein, which depends largely upon geometrical intuition. Theorems which serve as a means for determining the relatively difficult linear groups in more than two variables are presented in chap, iv, and are, in fact, made use of in 781 V 4O7 viii PREFACE the chapters on ternary and quaternary groups (chaps, v and vii). Concerning the theory of group characteristics (chap, vi), an attempt has been made to exhibit this sub ject in a very simple form, by means of explicit definitions and easy proofs which eliminate complicated sigma- constructions. The student is here (as elsewhere) urged to work with the matrix form of a linear transformation and intransitive group ( 2, 85). Where purely geometrical methods are not available or are less convenient than analytical methods, col- lineation groups are most easily studied through the medium of " linear groups" (cf. 10, 51); hence the almost exclusive discussion of the latter category in the text. The author owes his thanks to his colleagues, Profes sors R. E. Allardice and W. A. Manning, for the care with which they have read the manuscript and for many helpful criticisms. He is also deeply indebted to Pro fessors E. H. Moore and L. E. Dickson of the University of Chicago for their generous encouragement and valuable suggestions. H. F. BLICHFELDT LELAND STANFORD JUNIOR UNIVERSITY TABLE OF CONTENTS CHAPTER PAGE I. ELEMENTARY PROPERTIES OF LINEAR GROUPS .... 1 1-6, Linear transformations: definitions, funda mental properties. 7-14, Groups of linear trans formations: classification. 13, Change of variables. 14, Transitive and intransitive groups. 15-20, Hermitian invariant and reducibility of linear groups. 19, Unitary transformations. 20, Reducible and irreducible groups. 21-22, Canonical form of a linear transformation and of abelian groups. 23, Characteristic and chacteristic equation. II. GROUPS OF OPERATORS AND SUBSTITUTION GROUPS 29 24, Introduction. 25-39, Groups of operators. 25-26, Operators. 27-28, Finite groups; gen erators; subgroups. 29-31, Conjugate sets and subgroups; invariant subgroups; simple groups. 32-33, Isomorphism; factor groups. 34, Abe lian groups. 35-39, Groups whose orders are powers of a prime number. 36-38, Sy low s theorem. 40-45, Substitution groups. 40-42, Definitions; notation; even and odd substitutions. 43, Substitution groups; symmetric and alternating groups. 44-45, Transitive and intransitive groups; theorem. 46-47, On the representation of a group of operators as a substitution group; regular group. 48-50, On simple groups. 49-50, Theorems on the alternating group. III. THE LINEAR GROUPS IN Two VARIABLES 63 51, General remarks on linear groups and collineation groups; equivalence. 52-55, Determination of the linear groups in two variables. 56-58, The groups of the regular polyhedra. 59, Jordan s process: the diophantine equation, ix X TABLE OF CONTENTS CHAPTER PAGE IV. ADVANCED THEORY OF LINEAR GROUPS 76 60, Primitive and imprimitive groups. 61-62, On the form of Sylow subgroups; theorems. 63-74, On the order of primitive groups. 63-64, Superior limit to the magnitude of the prime factors. 65- 68, Superior limit to the factors which are powers of a prime number; theorem on two commutative trans formations. 69-73, Superior limit to the order of an abelian subgroup. 74, Superior limit to the order of a primitive group in n variables; historical note. V. THE LINEAR GROUPS IN THREE VARIABLES 104 75, Introduction. 76-77, The intransitive and im primitive groups. 78-79, Primitive groups hav ing invariant intransitive or imprimitive subgroups. 80-82, The primitive simple groups. 83, Primi tive groups having invariant primitive subgroups; bibliography. VI. THE THEORY OF GROUP CHARACTERISTICS 116 84-88, Introduction; definitions; the sum of matrices; invariants. 89-91, On the character istics of transitive groups. 92-94, On the char acteristics of isomorphic groups; composition of two groups. 95-99, On the totality of non-equivalent isomorphic groups; the regular group. 100, An application: no group of order p"q b can be simple. VII. THE LINEAR GROUPS IN FOUR VARIABLES 139 101 Introduction. 102-17, The primitive simple groups. 118-19, Groups which contain primi tive simple groups as invariant subgroups. 120- 25, Non-primitive groups and primitive groups which contain invariant non-primitive subgroups. TABLE OF CONTENTS xi CHAPTEK PAGE VIII. ON THE HISTORY AND APPLICATIONS OF LINEAR GROUPS 174 126, The history of linear groups. 127, Klein s ex tension of the Galois theory of equations. 128, The connection between linear differential equations hav ing algebraic solutions and linear groups. APPENDIX 183 129-31, On congruences and indeterminate equa tions. 132, On the value of a certain determinant. 133, On roots of unity. 134, On algebraic inte gers. INDEXES . 191 CHAPTER I ELEMENTARY PROPERTIES OF LINEAR GROUPS LINEAR TRANSFORMATIONS: FUNDAMENTAL PROPERTIES, 1-6 1. Examples of linear transformations. Operator. Let the axes of co-ordinates (rectangular) in the plane be rotated through an angle 0, the origin remaining fixed. If x, y are the co-ordinates of any given point with refer ence to the axes in their original position, and x , y the co-ordinates of the same point with reference to the axes in their new position, then, as is proved in analytic geometry, , , x = x cos y sin , y = x sin0-fy cos0. Hence, if a given curve has for its equation (2) /(*,) -0 with reference to the old axes, then its equation with reference to the new axes will be (3) f(x cos 0-y f sin 0, x sin B+y cos 0) = . We shall say that (3) is obtained from (2) by the linear transformation (1). In conformity with the general group terminology we call the transformation (1) an operator, which, operating upon (2), produces (3). To take a second example: any literal substitution (cf. chap, ii, B) can be exhibited as a linear transformation. For instance, the substitution (xix&) (x 4 x 5 ) can be written 2 FINITE COLLINEAT1ON GROUPS 2. Formal definition. A linear transformation in n variables is a set of linear homogeneous equations, (4) x 2 . . . +a nn x expressing the original variables x\, . . . , x n in terms of new variables x[, . . . , x,[, under the condition that the equations can be solved for the latter; that is, the determi nant of the coefficients a st (s, t= 1, 2, . . , n), called the determinant of the transformation, must not vanish. We represent a linear transformation by a capital letter (as A, S, . . ), or, if we wish to be specific, by the matrix of the linear transformation [an ai2 . . ai n ~\ a n i a nZ . . a nn \ which may be abbreviated to [a st \. Thus the equation A = [a st ] implies that a transformation denoted by A has for coefficients the numbers a st (s, t=l,2, . . , n). The same implication is also indicated by writing A: to the left of the equations (4) . The equation A = B is equiva lent to the statement that the matrices of A and B are identical. We do not a priori place any restrictions on the form of the numbers a s t- They may be real or imaginary, rational or irrational. The transformation (4), which we shall denote by A, operates upon a given function f(x\, . . , x n ) by putting in place of x\, . . , x n the equivalent expressions in the right-hand members of (4). This is indicated symboli- PROPERTIES OF LINEAR GROUPS 3 cally by (f)A. However, for reasons apparent later, we shall drop the accents after the operation has been per formed. Thus, if S is the transformation (1) and / the function (2), the left-hand member of the equation (3) will be written (f)S=f(x cosB-y sin 0, x sin 0+y cos 0) in the future. Certain special forms of linear transformations occur frequently or are introduced for convenience: (a) Canonical form. If every a st = in (4) except when s = t, we say that A has the canonical form. In this case we write A = (a u , #22, . . , a n ). The coefficients On, #22, . . , a nn are here called the multipliers of A. The transformations A 2, E, B^ B*, 7, all have the canonical form. (6) Similarity-transformation. This is a transforma tion in canonical form all of whose multipliers are equal (an = a 2 2= . . =a/m), as in the cases A 2 , E, 7. (c) The identity. This is a similars-transformation whose multipliers are unity. We shall reserve the letter E for this transformation: #=(1,1, . . ,1). Evidently, the identity produces no change; that is, (f)E=f. For an illustration, put = in (1). 3. The product of linear transformations. If we operate on a given function/ by two linear transformations successively, say first by A and then by B, the result, which is written (f)AB, is equivalent to that obtained by- operating on / by a single linear transformation C. For instance, let/ be a function f(x, y), A the transformation (1), 1, and B=(a, 6), and we find x - ^ y* (f)AB=f(xa cos B-yb sin 0, xa sin 0+yb cos 0), 4 FINITE COLLINEATION GROUPS which is obviously also the result of operating on/(z, y) by C: x = x a cos 6 y b sin 6, y = x a sin 0+y b cos 0. Remark. We notice that (f)BA =f(xa cos ya sin 0, xb sin 6-\-yb cos 0), so that (f)AB^(f)BA unless (a-b) sin = 0. THEOREM 1. LeZ A = [a 8t ] and B = [b 8t ] be two linear transformations in n variables. Operating first by A and then upon the result by B is equivalent to operating originally by a single linear transformation C = [c st ] in n variables, where +a sn b nt We say that C is the product of A and B, and write symboli cally AB=C* The rule for finding the product AB = C may be described in the following manner (matrix multiplication). We define the product of a row (horizontal) of A, say 2 i, 22, . - , azn, and a column (vertical) of B, say 613, 623, - - , bm as follows: multiply the first element of the row by the first element of the column, the second element of the row by the second element of the column, etc., and add the resulting n products (021613+022623+ . . +2n6/t3 in the * A value for c st different from that given above is obtained by writing last in the product that transformation which operates first, as is the custom with functional operators (BA(f)), or by regarding the accented letters in (4), 2, as the old variables and the unaccented as the new. Thus, Klein, Jordan, Burnside, and Weber obtain _,* = n c st = Z k = l **** while Schur and Probenius get the value given in (5). The author has hitherto (in papers published in technical journals) accented the old and left unaccented the new variables, and has used the term "linear substitu tion" for "linear transformation." PROPERTIES OF LINEAR GROUPS 5 example). Then c s t is the product of the sth row of A and the tth column of B. The value of c st given above is found by eliminating x(, x z , . . , x n from the two sets of equations A: B: x s = b sl x f l + . . . +b sn x f ^ (s = l,2, . . , n). 4. The commutative and associative laws. Power of a linear transformation. In general, the commuta tive law does not hold; that is, AB differs from BA (cf. Remark, 3). On the other hand, the associative law holds in a product of three or more transformations. Thus, let A, B, C be any three transformations, and let AB = P and BC = Q. Then (AB)C = PC = AQ = A(BC). As a consequence the notation ABC is not ambiguous, and we shall write A 2 for AA, A 3 for A 2 A, etc. We call A m the rath power of the transformation A. To prove the associative law, we construct the matrices of p = [p st ]=AB, Q = [q s t]=BC by Theorem 1, 3, and then those of PC and AQ. Linear transformations having the canonical form are commutative and their products are readily written down. In particular, if S=(a n , 22, . . , a nn ), T=(b n , 6 22 , . . , b nn ), we have ST=TS=(a u bn, a 22 6 22 , . . , a nn b nn ), and 5. The inverse of a linear transformation S is such a transformation, written S~ l , that the product SS~ 1 is equivalent to the identity E; i.e., produces no final change. In the case of the transformation (1), 1, the inverse is 6 FINITE COLLINEATION GROUPS plainly the rotation of the axes through the angle 0: x = x cos B-\-y f sin 0, y = x sin 6-\-y f cos 0. It is also plain that, having transformed a function f(x, y) by means of (1) into a function F(x , y ), we get the original function f(x, y) by substituting in F the values of x , y expressed in terms of x, y. These solutions appear in the form of a linear transformation after the accents have been properly placed. THEOREM 2. The inverse of a linear transformation S: x s = a s ix + . . +a sn Xn (s = l, 2, . . , w), is obtained by solving this system of linear equations for xl, x(, . . , x n in terms of Xi, x 2 , . . , x n . After the accents have been properly placed, these solutions appear in the standard form of a linear transformation, denoted by S~ l . More generally, we shall denote the inverse of S m by S~ m , and we have S m S- m = S- m S m = E. We can now prove that, given AB = AC, then B = C. For, from A~ 1 (AB)=A~ l (AC) follows (4): Similarly, if BA = CA, then B = C. 6. Order of a linear transformation. Consider the transformation (1), 1, which we shall indicate by S. If 6 is an aliquot part of 2?r, then some power of this transformation will be equivalent to the identity. Thus, if is a right angle, S* = E. In general, if S is an arbi trarily chosen transformation, the identity will not be found among the powers of S. If it should be, we say that S is of finite order; and if m is the lowest positive integer for which ti m = E, we say that S is of order m. In this case no power of S need be taken higher than m. PROPERTIES OF LINEAR GROUPS 7 For, since S m+a = S m S a , and the operator S m produces no change, it follows that S m+a = S a . If S is of order m, the order of S h is m/d, where d is the highest common factor of ra and k (cf. 26, (6)). Thus, if m=12, then S 5 , S 7 and S 11 are of order 12; S 4 and >S 8 of order 3, and so on. EXERCISES 1 . If di and d 2 are the determinants of the linear transformations A i and A z in two variables, then the determinant of A\A Z is d\d 2 . 2. Prove that the product of a linear transformation T and a similarity-transformation S = (a, a, . . , a) is obtained by merely multiplying every element in the matrix of T by a. Hence prove that a similarity-transformation is commutative with every transformation in the same variables. 3. Find the general form of a linear transformation in three variables which is commutative with S = (a, a, 6). 4. Prove that the two transformations are each the inverse of the other. 5. If S is of order m, then S m ~~ 1 is the inverse of S. 6. A linear transformation is the inverse of its own inverse. 7. Are the following transformations of finite order: en- 8. Prove that the multipliers of a transformation of finite order are roots of unity (cf. 133). GROUPS OF LINEAR TRANSFORMATIONS I CLASSIFICATION, CHANGE OF VARIABLES, 7-14 7. Finite groups of linear transformations. Let there be given a set of distinct linear transformations, finite in number, and let it be known that the product of any two 8 FINITE COLLINEATION GROUPS of the set (AB as well as BA), whether alike or distinct, is again a transformation of the set; then this set is called a finite group of linear transformations. We shall usually employ the simpler expression linear group. The number of distinct transformations in the set (including the iden tity) is called the order of the group. As an example of such a group take the four trans formations consisting of the rotations of the X , Y axes through 1, 2, 3, and 4 right angles around the origin in their plane: -n r-i on r o 11 1 oj A *=\ o -ij> M-i oj> -B a- For a second example, take the eight transformations consisting of the four rotations just given, in addition to these four accompanied by a reflexion on the new Y axis: Ai, A 2 , A 3 , E; -n a- The former group is a subgroup of the latter. The set of three transformations A i, A 2 , A z do not form a group, since the transformation A\ (or AI A 3 ) is not found in the set. Notation. We shall reserve the letters G, H, K to denote particular groups that may come under discussion. The equation G = (Si, Sz, . . , S g ) implies that a given PROPERTIES OF LINEAR GROUPS 9 group G consists of (or contains) the transformations S\ t S2, . . , Sg. It is often convenient to use the phrase " (a group) G" with the same meaning as the phrase "the transformations of (a group) G." For instance, we may say "(something) is unaltered by G" instead of "(something) is unaltered by the transformations of G." 8. Elementary properties of linear groups. Gen erators. Let G represent a given linear group, and S any transformation contained in G. Then it is easy to show that S is of finite order, say ra, and that its different m powers (including its inverse and the identity) are con tained in G. For, the series of transformations S, SS = S 2 , SS* = S*, . . . all belong to G and can evidently not form an unlimited number of distinct transformations, the group being finite. Hence, at least two of these powers are equivalent, say /S = 0+m , which may be written S a E=S a S m . It fol lows that E=S m (5). The transformation S is there fore of finite order (6), and /S" 1 " 1 is its inverse. It may happen that the various powers of S exhaust the transformations of G. We then say that G is generated \ by S, and that S is a generator of G. If G contains otner transformations, let T be one such, and the products S a T b , S a T b S c , ... all belong to G. We may be able to get all the transformations of G in this manner; if so, we say that S and T generate G, or that they form a set of generators of G; and so on. Consider for example the group (6), 7. Here A\ = AZ, Al = A 3 , A\ = E. Hence AI generates this group, which is also geometrically evident. In the case of (7) we have additional relations A iBi = B 2 , A\Bi = Bz, A\B\ = B^ so that AI and BI form a set of generators for this group. 10 FINITE COLLINEATION GROUPS Having given a linear group, a set of generators may usually be selected in different ways. It should be noted that a linear transformation selected at random will not generate a finite group (6), and that two or more linear transformations each of finite order, but otherwise taken at random, will not generate a finite group. 9. Collineations and collineation groups. If x\, . . , x n represent homogeneous co-ordinates in space of n 1 dimensions, then the geometrical effect of a linear trans formation is not altered by multiplying all the elements in its matrix by an arbitrary constant. To illustrate, if Xi, ;, x 3 represent trilinear co-ordinates of the plane, the transformations A = are both equivalent to the projective transformation leaving fixed the straight lines and transforming the point (1, 2, 3) into the point (1, 0, 1). We say that A and A represent the an,me collineation; that is, a collineation is specified by the mutual ratios of the elements in the corresponding matrix, not by the actual values of these elements (to a given non- vanishing element may therefore at the outset be assigned at will any convenient number not zero). In practice it is customary to affix a factor of proportionality to either the old or the new variables to distinguish a collineation from a linear transformation; the collineation represented by the two transformations above will thus be written PROPERTIES OF LINEAR GROUPS 11 The following laws obtain : (a) If A and A are linear transformations represent ing the same collineation, then there is a similarity- transformation S such that A = AS = SA (or A A~ l = A~ 1 A = S). Thus, in the example above, =(2, 2, 2). Conversely, S being any similarity-transformation, A and AS represent the same collineation. (6) If A and A represent the same collineation Ci, and similarly B and B represent the same collineation C 2 , then A B and A B represent the same collineation CiC,. For, let S and T be the two similarity-transformations A A- 1 and B B~\ so that A = AS, B = BT, and we get A B = ASBT = ABST, since S is commutative with B ( 6, Exercise 2). Hence, ( : 4B)- 1 (A J B ) = (A J B)- 1 (^jBAS7 7 ) = (A J B)- 1 (^^)(57 1 )=/S7 7 , which is a similarity-transformation (4). Hence, to find the product of two collineations, C\ and C 2 , we take the product of any two representative linear transformations. If therefore we have a finite set of collineations such that the product of any two of the set (whether alike or different) is a collineation of the set, we call this set a collineation group whose order is the number of distinct collineations of the set. 10. The collineation group derived from a given linear group. Let G be a linear group of order g, and A any one of its transformations. Furthermore, let K=(Si, 82, . . , Sk) be the group consisting of all the similarity-transformations contained in G ( 11, Exercise 1). Then (9, (a)) ASi, AS 2 , . . , AS k are distinct linear transformations, all representing the same collineation, say C". Moreover, no further trans formations of G can represent C . 12 FINITE COLLINEATION GROUPS If now B is a new transformation of G, we get a new collineation C" corresponding to the transformations Proceeding thus, we shall finally arrange all the g trans formations in g/k classes, giving rise to a set of g/k distinct collineations. These form a group of order g/k, since the product of any two of them belongs to the set (cf. 9, (6)). We formulate this result as follows: THEOREM 3. To a given linear group G of order g corresponds a collineation group G f of order g/k, where k is the order of the group of similarity-transformations K contained in G. To a given collineation correspond k linear transformations of G, obtained from one of them by multiplying it in turn by each of the transformations of K. Example. Let G be the group (6), 7, of order 4. Here K=(A 2 , E), and we have two collineations in G , represented respectively by (A 2 , E) and (Ai, A 3 ): E : px= x , py = y f ; A{: px=-y , py = x . If a linear group G of order g contains no similarity- transformation other than the identity, then G will itself represent the corresponding collineation group. In other cases it may, or may not, contain a subgroup of linear transformations of order g/k which represents the col lineation group corresponding to G. For instance, there is no linear group of order 2 contained in (6) whose collineation group is that one given in the example above. On the other harud, the group E=(l, 1), A 2 =( 1, 1), 2 =(1, 1), # 4 =( 1, 1) contains a subgroup which may be taken as its collineation group, namely E, J5 2 . PROPERTIES OF LINEAR GROUPS 13 11. Linear fractional groups. A collineation in n variables Xi, . . , x n may be represented as a linear fractional transformation in the n-\ ratios yi = xi/x n , y 2 = x 2 /x n ,. . ,y n -i = x n -i/x n . Assum ing for simplicity n = 3, the collineation corresponding to the linear transformation as a transformation in the variable yi, y 2 , takes the form 2/1 auy( ~ a n y[+ a 32 y 2 +a 33 The transformations of K (cf . 10) will all become the identity and the k transformations representing a single collineation will give rise to a single linear fractional transformation. We thus obtain a linear fractional group of order g/k which is simply iso- morphic (cf . 32) with the collineation group G and may be regarded as its equivalent. EXERCISES 1. Prove that, in a linear group G, those transformations which have the canonical form make up a group by themselves. More particularly, the similarity-transformations make up a group K which is invariant under G (cf. 31). 2. If G and G are a linear group and its corresponding collineation group, then to the identity of G correspond all the similarity-transformations of G. 3. From the formulas for the product of two linear transforma tions (3) and the product of two determinants, prove that the product of the determinants of two transformations A and B is equal to the determinant of the transformation AB (cf. Exercise 1, 6). 4. The determinant of the linear transformation A is the rath power of the determinant of A (cf. Exercise 3). Hence prove that the determinant of a transformation belonging to a linear group is a root of unity (cf. 133). In particular, the determinant of a transformation of the third order is 1, w or w 2 , where w 3 = l. 5. Construct the collineation group corresponding to the group (7), 7. Show that there is no subgroup of (7) of order 4 which may represent this collineation group. 14 FINITE COLLINEATION GROUPS 12. Groups of linear transformations of determinant unity. The problem, having given a collineation group G of order g in n variables, to construct a corresponding linear group G, admits of an unlimited number of solutions (cf. 9, (a)). We shall limit the problem by requiring the determinants of the linear transformations of G to be unity, and we shall show that under this condition the order of G is not greater than ng r . Let A be a linear transformation representing one of the collineations of G , and let its determinant be d. We then multiply it in turn by each of the n similarity- transformations Si, Sz, . . , S n , where Sj=( r j> TJ, . . , TJ), r\, 7*2, . . , r n being the n different roots of the equation r n d=l. The n transformations so obtained all have a determinant = 1 ; moreover, no linear transformation outside these n will be of determinant unity and will represent the same collineation as A ( 9, (a)). Taking each of the g r collineations in turn, we shall have constructed a table like that in 10, containing ng linear transformations in all. If A , B r , C are three collineations in G such that A B = C , then will the prod uct of any transformation of our table from the line corresponding to A and any transformation from the line corresponding to B f , necessarily be a transformation from the line corresponding to C", since the determinant of this product is unity (Exercise 3, 11). In other words, the ng linear transformations form a linear group G, whose collineation group is G . Example. Let = (#, A), where A = (l, -1). From E we obtain the two transformations E, Ei = (l, 1); and from A the two AI = ( i, i), A 2 =(i, i), where i=V-\. Therefore G=(E, E ly Ai, A 2 ). PROPERTIES OF LINEAR GROUPS 15 It may happen that a group Gi of transformations of determinant unity exists whose order is lower than ng and whose collineation group is likewise G f . Its order must be divisible by g and be a divisor of ng ( 10, 28). EXERCISES 1. Construct a group of order 3 of linear transformations of determinant unity whose collineation group is (1, 1), (1, ), (1, <*?); o 3 = l. 2. Construct the group of order 12 of transformations of deter minant unity whose collineation group is r-i 11 ro n r o n r-i n r-i 01 ""I o J; L oj U ij- U 0} l-rij: (As a linear fractional group, this group has the form y=y , -y +i, W, i/(-V 13. Change of variables. For certain purposes it is of great convenience to introduce new variables which are linear functions of the old. To illustrate the theory let us consider the transformation (1), 1: S: x x cos y sin 0, y = x f sin 0+y cos 0. We shall now introduce new variables X, Y, where (8) X = x+iy, Y =x-iy (i=l/^I), and correspondingly /0/\ ~Y f /r J-Vi/ V T <iii f (o ) A x -\-iy y i x ly . The transformation S becomes S : X=e i& X , Y=e- ie Y , a result that can be expressed symbolically in terms of S and the change of variables (8), which we shall regard as a linear transformation. 16 FINITE COLLINEATION GROUPS Solving (8) for x, y we obtain T: x=(X+Y)/2, y=(X-Y)/2i. Consider an arbitrary function f(x, y). The trans formation S changes / into the function f(x cos 6-y sin 0, x sin 6+y cos 0), which, expressed in terms of the variables X , Y , is the function X +Y X -Y . , X +Y . .X -Y A ^ cos0 -r- sm0, - sm0+ 2i cos0). On the other hand, we may at the outset express / as a function of the new variables: f((X+Y)/2, (X-Y)/2i), and then transform it by means of S : e i9 X +e~ i9 Y e i9 X -e- i9 Y \ 2 2i ) Symbolically stated, the operator ST is equivalent to the operator TS : (f)ST=(f)TS f . Hence, We formulate this result as follows for the general case : THEOREM 4. Having given a set of linear transforma tions A, B, . . and a function f, involving n variables Xi, . . , x n , we may introduce new variables by means of a linear transformation T: Xft = feiXi+fe2X 2 + - +fenX n (&=!, 2, . . , n), changing f into a function of Xi, . . , X n . As linear transformations in the new variables, A, B, . . will take theformsT~ l AT, T~ 1 BT, . . . PROPERTIES OF LINEAR GROUPS 17 We note that if AB = C, then (T- 1 AT)(T~ 1 BT) = T~ 1 CT. Hence, if A, B, . . form a group, so do T~ 1 AT, t-^BT, . . , and the two groups are simply isomorphic ( 32). 14. Transitive and intransitive groups. Groups in twq, three, and four variables whose transformations have respectively the following forms: a b c Ode p q r s t u .0 v w p a b c d e f -0 g h j. are said to be intransitive. It may happen that a group can be made to assume the intransitive form by a suitable change of variables, though it does not possess this form initially. Consider, for example, a group G in four variables x\, . . , x 4 whose transformations are all of the following type: a b e f b a f e h g c d g h d c] The introduction of new variables y\, y 2 , z\ y z 2 , where yi = Xi+x 2 , y 2 = x 3 +Xi; Zi = xi-x 2 , z 2 = x z -x 4 , will change these matrices into the type C\ above. That is, G has the intransitive form when written as a group in the variables yi, y 2 , 2i, z*. DEFINITION. If the n variables of a group G can be separated into two or more sets (either directly or after a suitable change of variables), such that the variables of 18 FINITE COLLINEATION GROUPS any one set are transformed by all the transformations of G into linear functions of the variables of that set only, we say that G is intransitive. If such a division is not possible, the group is transitive. The different sets into which the variables of an intransitive group may be separated (as the sets (y 1} y 2 ) and (zi, z 2 ) of the group above) are called its sets of intransitivity. EXERCISES 1. Prove that a change of variables will not alter the form of a similarity-transformation. 2. Prove that ^ = [Q ] can be obtained from A = f a ] by a suitable change of variables if a *6, and find the correspond ing transformation T. (Hint: The condition T~ 1 AT = A gives AT = TA . Multiply out and determine the elements in the matrix of T from the four resulting equations.) 3. Prove that if in type Ci, t = p, u = q, v=r and w = s, then the two sets of intransitivity can be chosen in an infinite number of ways. HERMITIAN INVARIANT AND REDUCTIBILITY OF LINEAR GROUPS, 15-20 15. Conjugate-imaginary groups. Let G=(A, B, C, . . ) be a linear group in the variables x\, . . , x n , and assume that the elements in the matrices of the various transformations are not all real. We may then separate the real and imaginary parts in both variables and coefficients; and by passing to the conjugate- imaginary values we evidently obtain a new group G=(A, B, C, . . ) in the variables x\, . . , x n (we shall denote the conjugate-imaginary of a quantity w by w), simply isomorphic with G (32). For, if AB = C, it follows that AB = C. We shall call either group the conjugate-imaginary of the other. PROPERTIES OF LINEAR GROUPS 19 16. Hermitian form. The expression k=l 1=1 subject to the condition that it vanishes only if Xl = x 2 = . . = x n = 0, and is real and positive for all other sets of values that we may assign to these variables, is called a positive-definite Hermitian form in the n variables x lf . j x nj or simply Hermitian form. For instance, the expression XiXi+3x 2 X2+(l-}-i)xiX2+(l i)x 2 Xi is a Her mitian form in the variables x\, x 2 . THEOREM 5. A positive-definite Hermitian form J in the variables x\, . . , x n may be reduced to the form 2/12/1 + 2/22/2+ + 2/n2/n by a change of variables of the following type: 2/2 = Proof. Arranging J according to x n and x n we have J = Jn = QnnXnXn + ^ n X n _ i + X n X n _ i + X, where X n -i represents a linear function of x n -\, . . , x\. The coefficient q nn is real and positive, since it is the value of J obtained by putting x n =l, x n - } = x n - 2 = . =zi = 0. Accordingly, if we put + l/g^=+l/^ = pnn, and X n _i = p nn 7 n _i, we have Jn=(pnnXn+ Y n -l) (pnnX n -l+ ?n-l) +X - Y n .,Y n ^ 20 FINITE COLLINEATION GROUPS where J n -\ is a Hermitian form in n 1 variables x n -i, . . , Xi. For, it is of the required type and is the value of J n obtained by subjecting the variables to the single condition x n = Y n -i/p nn . It is therefore real and positive unless x n -\= . . = Xi = 0. We now arrange J n _ x according to x n -i and z n _i, and proceed as above. We find where y n -i is a linear function of z n _i, z n _ 2 , . . , *i. Continuing thus, we finally prove the theorem. 17. Lemma. // G=(T lf T 2 . . . , T g ) is a linear group, and f any function of the variables of the group, then the function 0) 7=(/)r,+(/)r,+ . . +(j)T, is either identically zero or is an "invariant" of G; that is, it is transformed into itself by every transformation of G. Proof. We have (10) (^ k =(f)T 1 T k +(f)T t T k + . . + (f)T T k = (/)+(/)+ . +(f!M, say. But, T[, T, . . , TO are the transformations Ti, T 2 , . . , T g over again in some order, since they all belong to G ( 7) and are all distinct ( 5; cf. Exercise 3, 27). It follows that the last sum of (10) is equal to the right-hand member of (9); i.e., I=(I)T k . 18. Invariant Hermitian form. We say that a Hermitian form J is invariant under a group G, or that J is n. J1p.rmH.ian invariant of G, when J is transformed into itself by the transformations of the intransitive^ group made up of G and its conjugate-imaginary group G. PROPERTIES OF LINEAR GROUPS 21 THEOREM 6. There is always a Hermitian invariant J of a given linear group G in n variables* Proof. Let the transformations of the group made up of G and G be denoted by T it T 2 , . . , T , and let / represent the function XiXi+x z X2-\- . . +x n x n . Then the function . . + (I)T g is the required Hermitian invariant. First, J is a Hermitian form in the variables x\, . . , x n . For every term (I)T k is the sum of n expressions (x a x a )T k = X s X s , each of which is the product of two conjugate-imaginary quantities. The function J is there fore real and non-negative, and cannot vanish unless every term (I)T k vanishes. But if Ti represents the identity, (I)Ti = I and does not vanish unless every variable x\, . . , x n vanishes. This is therefore also the case with J. Second, J is transformed into itself by TI, . . , T n , by the lemma, 17. By aid of the theorems 5 and 6 we derive the COROLLARY. Such variables x\, . . , x n may be selected for a linear group G that the function is a Hermitian invariant of G. 19. Unitary transformations. A linear transformation A = [a st ] whose coefficients satisfy the following relations : (11) ai k fli h +o 2 ifl2k+ +a nk a nk =l (fc = l,2, . - , n), * This theorem was proved for n =3 by Picard and Valentiner (1887, 1889), and for any n by Puchs, Loewy, and Moore (189G). See Encyclo pedic der mathematischen Wissenschaften, Leipzig, 1898-1904, Bd. I, 1, p. 532. 22 FINITE COLLINEATION GROUPS (11 ) t ti*0iz+02*o 2 z + . . +o*a n /=0 (12) a *io*i+ 0*2^*2+ - . -\-a kn a kn =l (12 ) a is said (k,l = l,2, . .-, to have the unitary form. n; fc=H), The variables of a group being selected so that its Hermitian invariant is XiX\-\- . . +x n x n , we readily find that the corollary of 18 is tantamount to the follow ing statement : such variables may be selected for a linear group that its transformations all have the unitary form. The equations (11), (11 ) are deduced directly; and the equa tions (12), (12 ) by operating on the Hermitian form by A~ l , as given below. The inverse of a transformation in unitary form can be written down at once: A- 1 : x k = ai k x 1 +a 2k x 2 + . . +a nk xl l (k = l, 2, . . , n). For, the condition A~ 1 A = (l, 1, . . , 1) leads to the equations (11), (!! ) 20. Reducible and irreducible groups. A linear group in n variables is said to be reducible when, after a suitable choice of variables x\, . . , x n , a certain number of these (say Xi, z 2 , . . , x m ; m<n) are transformed into linear functions of themselves by the transforma tions of the group. (Thus, a group in three variables Xi, x z , z 3 , whose matrices have the form PROPERTIES OF LINEAR GROUPS 23 is reducible.) We shall say that the m variables x\ t . . , x m constitute a reduced set of the group. An irreducible group is a group in which no such choice of variables is possible. THEOREM 7. A reducible group G is intransitive, and a reduced set becomes a set of intransitivity. Applied to the reducible group in three variables indicated above, the theorem asserts that new variables may be introduced such that the matrices are of type B, 14, and that the original variables x\, x 2 will form a set of intransitivity of the new. Proof. A Hermitian invariant of G may be reduced to the form y\y\-\- . . -\-y n y n by the change of variables specified in 16. The group G will still have the typical form of a reducible group, whose matrices we shall write symbolically \A LA" A "\ and it remains for us to prove that the elements in A " are all zero. For this purpose we write down as many of the equa tions (12) and as many of the equations (11) as contain elements of A "; namely the last n-m in each case. If we then subtract the sum of the latter equations from the sum of the former, there results an equation 2a s< a s< = 0, in which the left-hand member contains one term for each of the elements of A". Now, each of these terms is real and non-negative; consequently the sum 2<a st a st cannot vanish unless every number a s( = 0. That is, the elements of A" are all zero, and the theorem is proved. The validity of the proof is evidently not impaired by assuming the given reducible group G to be a subgroup of a larger group H, reducible or irreducible. Furthermore, the final intransitive group G is composed of unitary 24 FINITE COLLINEATION GROUPS transformations, by the process of proof; and this result is equally true of H (let the initial Hermitian invariant used in the proof be the Hermitian invariant of H) . Thus, with slight modifications of the above proof as to detail, which will be left as an exercise for the student, we obtain the following important THEOREM 8. Having given a linear group H contain ing an intransitive subgroup G, we may choose such variables that G appears in intransitive form (cf. types A-C, 14), and that at the same time the transformations of H are all unitary. EXERCISES 1. Show that a unitary transformation in two variables and of determinant unity has the form pp+qq = l . 2. Prove by the method of 18 that if all the elements in the matrices of G are real, then there is a quadratic function of the variables which is invariant under G. 3. Find the most general type of a Hermitian invariant of a linear group which contains a transformation in canonical form whose multipliers are all distinct, as (1, w, w 2 ); 3 = 1. 4. Prove that if a group G possesses a Hermitian invariant which does not contain all the variables of G, then this group is intransitive. 5. If there exists a linear function of the variables of a group G which is invariant under G, then this group is intransitive. In the case of a substitution group (chap, ii) written as a linear group (1) there is such a function, namely the sum of the letters of substitution. Hence this group is always intransitive. CANONICAL FORM OF A LINEAR TRANSFORMATION AND OF ABELIAN GROUPS, 21-22 21. Theorem 9. A linear transformation of finite order will assume the canonical form by a suitable choice of variables. PROPERTIES OF LINEAR GROUPS 25 Proof. Let the transformation be A = [a st \. We can determine a linear function which is transformed into a certain constant multiple of itself by A, say (y\)A=Oy^ where y\ = biXi+ . . +b n x n . To obtain the necessary conditions we equate the coefficients of the variables Zi, . . , x n , and find the equations biait+b2flu+ . . +b n a nt = 0b t (*=1, 2, . . , n), which can be solved for 61, . . , b n provided is a root of the characteristic equation of A (cf. 23). Having thus determined yi We change to new variables such that y\ is one of these. The group generated (8) by A is now seen to be reducible, since from which it follows that the first row in the matrices of each of these transformations is of the form k . .0, where k represents different powers of in the different transformations. Hence, by Theorem 7, 20, the group in question is intransitive, and yi constitutes one of its sets of intransitivity. Let (yz, . . , y n ) be the other (temporary) set of intransitivity. The foregoing process may now be repeated for the set (2/2, , 2/ n ) in place of the original n variables. A new linear function will be determined which is transformed into a constant multiple of itself by A, and the set (2/2, , y n ) will break up into further sets of intransi tivity. Continuing thus, we finally obtain A in the desired canonical form. 22. Canonical form of abelian group. The group consisting of the different powers of a transformation A 26 FINITE COLLINEATION GROUPS is an example of a type of group called abelian; namely a group in which all the operators are commutative (4); i.e., if A and B are two operators of the group, then AB = BA. THEOREM 10. In any given abelian group K of linear transformations, such new variables may be introduced that all the transformations of K will simultaneously have the canonical form. Proof. If the group contains only similarity- transformations (2) the theorem is self-evident. Hence we assume in K a transformation S which is not a similarity-transformation. Let the variables of the group be chosen such that S appears in the canonical form ( 21) : s=(, ..,*)"; The multipliers of S may not all be distinct. Sup pose that m of them are equal (say to a), and differ in value from all the others; we shall then show that K transforms the corresponding variables into linear func tions of themselves and is therefore intransitive ( 20). Let / be any linear function of the m variables in ques tion, say Xi, Xz, . . , x m . We have (f)S = af; moreover, any linear function F such that (F)S = aF can evidently not contain any of the variables x m +ij . . , x n . Let now T be any transformation of K. We have ST=TS, and if we put (f)T=f we get That is, the function / , which is linear in the variables of the group, must be a function of x\, . . , x m only, by what has just been said. It follows that K is intransi tive. PROPERTIES OF LINEAR GROUPS 27 If we now confine our attention to one of its sets of intransitivity, we may apply anew the process above to that set. This will, therefore, break up into further sets of intransitivity. Continuing thus, the ultimate sets of intransitivity will contain one variable each, and the theorem is proved. We shall often say: "let a (given) transformation (or group) be written in canonical form" instead of "let the variables be so chosen that a (given) transformation (or group) will appear in the canonical form." From the theorems 8 and 10 we deduce the COROLLARY. Such variables may be selected for the variables of a linear group G that a given abelian subgroup of G is written in the canonical form, and that at the same time the transformations of G are all unitary. 23. Characteristic and characteristic equation. If we add to each of the elements in the principal diagonal of the matrix of a linear transformation A = [a st ] and equate to zero the resulting determinant, we have an equation in which is called the characteristic equation of A: (13) On ^ Oi2 021 022 ,-e = THEOREM 11. If T and A are linear transformations, the roots of the characteristic equation of A are the same as those of T~ l AT. Proof. Put T- 1 AT = B = [b st ], whose characteristic equation is bii 0. . bi n (14) =0. b n i . . b nn -0 28 FINITE COLLINEATION GROUPS Regarding as a variable temporarily, we shall look upon the left-hand members of (13) and (14) as the matrices of linear transformations which we shall write symbolically (A S) and (B S), where S repre sents the similarity-transformation (0, . . , 0). Then, since T~ 1 AT = B and T~ 1 ST = S ) we readily find that T~ l (A -S)T= (B-S). Hence, if the determinants of T, (AS) and (B S) are denoted by t, a, and b respec tively, we have (cf. Exercise 3, 11) t 1 at = b, giving a = b. Accordingly, the coefficients of the various powers of 6 in a and b are equal, and the theorem follows. The sum of the characteristic roots of A is called the characteristic of A. It is equal to the sum of the elements in the principal diagonal of A, namely 011+022+ . +nn- EXERCISES 1. Find the characteristic roots of a transformation written in canonical form. 2. Prove that the characteristic roots of a linear transformation of finite order are roots of unity (cf. 4, 6, 21, 23, 133). r i i 3. Can the variables be so changed that S = will assume the canonical form ? CHAPTER II GROUPS OF OPERATORS AND SUBSTITUTION GROUPS A. GROUPS OF OPERATORS 24. Introduction. The notion of a group was intro duced in 7. While this term has been associated hitherto with linear transformations only, there are so many important properties of groups of operators which are independent of their mode of representation, that it seems best to study such properties apart from the form that these groups take. The usefulness of the results derived will in this way not be limited to the realm of linear groups. In addition we need a certain amount of knowledge of substitution groups for the development of linear groups. However, beyond a general introduction to the theory of groups of operators and substitution groups, only such additional theorems in either field as are needed for this development will be given here. For a detailed account in the English language of abstract and substitu tion groups, the reader may with profit consult the follow ing books: E. Netto, The theory of Substitutions and Its Applications to Algebra (tr. by F. N. Cole). Ann Arbor, Mich.: Inland Press, 1892. W. Burnside, Theory of Groups of Finite Order. 2d ed. Cam bridge University Press, 1911. H. Hilton, An Introduction to the Theory of Groups of Finite Order. Oxford, 1908. G. A. Miller, H. F. Blichfeldt, and L. E. Dickson, Theory and Applications of Finite Groups. New York: John Wiley & Sons, 1916. 29 30 FINITE COLLINEATION GROUPS OPERATORS AND GROUPS OF OPERATORS, 25~28 25. Operators.* We postulate a certain class O of objects called operators having the following properties: 1. If A and B are any two operators, then either A and B are equal (A=B) or distinct (A^B). That is, the operators are so defined that it shall be possible for us in any case to determine whether two given operators are equal or not. Thus, if O is the class of all linear transformations in two variables, two operators A and B, defined by their matrices 3 Lc dJ -C are equal when the following (ordinary) equations are satisfied: a = p, b = q, c = r, d=s; if O is the class of all collineations in two variables, the, two corresponding operators are equal if a number k can be found such that the equations a = pk, b qk, c = rk, d = sk are satisfied. 2. Whether A =B or A 4=#, there is a unique operator C called the product of A and B; in symbols AB = C. Here we assume that a certain rule for forming the product AB (cf . 3) is given, producing an operator C in O, which is "unique" in the sense that if more than one operator results from the rule, then any two such operators are equal (cf. 9). 3. The associative law holds for a product of three operators: (AB)C = A(BC}\ that is, if AB = S, BC = T, 4. There is a unique operator E called the identity and having the property that, for every operator A, we have AE = A and EA =A. 5. The operators are reversible in 0; that is, to every operator A there corresponds a unique operator in O * The development of chap, i is followed closely in 25-27. For a bibliography and discussion of various definitions of abstract groups con sult E. V. Huntington, Transactions of the American Mathematical Society, VI (1905), 181 ft*. The postulates 4 and 5 above demand more than is logically necessary (cf. L. E. Dickson, Transactions of the American Mathe matical Society, ibid., p. 199). GROUPS OF OPERATORS 31 called the inverse of A and denoted by A" 1 , such that we In 4 and 5 the word "unique" is interpreted as in 2. 6. The following relations of equality obtain, as in the case of ordinary (numerical) equality: (a) A=A; (6) if A=B, then B = A; (c) if A=B and B = C, then A = C; (d) if A = B and C = D, then AC = BD. Remark. Our conception of the class of operators O involves necessarily another class F, composed of subjects of operation. To give some illustrations: (a) let F represent all polynomials in n variables and O all linear transformations in those variables; (6) let F represent all points in the plane and O all rotations in that plane around a given point, accompanied or not by inversions with respect to the given point; (c) let F represent n points on a line and O the different permutations of those points. However, in the present chapter the class F is practically never referred to. The product AB is not necessarily the same as the product BA. If the two products are equal (AB = BA), we say that the operators A and B are commutative. A continued product of any number of operators A, B, C, D, . . may be obtained by taking the product of two of them (say AB), then the product of this product and an operator, etc., giving say (((AB)C)D) . . . . By 3 it follows that the factors in the final product may be reassociated in any manner (as (AB)(CD) . . ), so long as their order in the product is not disturbed. In the future we shall often say "a (set, group, class) G contains an operator S," "S is found among the operators of G," "S is an operator of G," or simply "S belongs to G," instead of "S is equal to one of the operators of G. )} EXERCISES 1. Prove that the inverse of the identity is the identity (i.e., E~ l =E), and that the inverse of A~ l is A (i.e., A~ l A=E). (Hint: let A be the inverse of A~ l , then A~ 1 A = E. Now apply 3 to the left-hand member of A(A~ 1 A )=AE.) 32 FINITE COLLINEATION GROUPS 2. Prove that, if AB = AC, or if BA =CA, then B = C. 3. Prove that (S~ 1 AS) (S- 1 BS)=S~ 1 (AB)S. , 4. Let the operators AI, A 2 , . . of a class O be represented by the symbols (x\, yi), (x 2 , y 2 ), . . , under the condition that two operators are equal (Ai = A 2 ) only if the two equations Xi = x 2 , t/i = i/2 are satisfied. Now if the product A \A 2 is expressed by the formula 2, 2/2) = (x , y ), where x = XiX 2 yiy 2 , y = x\yi +y\x<L, prove that the conditions 1 6 are fulfilled, and find the identity and the inverse of A\. (To give an example of an operator of this type: let operating by A! con sist in multiplying by the complex number Xi+V li/i.) If the product is (x y y } = (xiX 2 , Xiy 2 ), prove that 1, 2, 3, 6 are fulfilled. 5. Prove that the inverse of the operator AB is B~ 1 A~ 1 . 26. Power and order of an operator. As in 4, we write A 2 for A A, A 3 for A (A 2 ) or (A 2 ) A, . . , A m for A(A m ~ l ) or (A m ~ 1 )A ) and call these products the 2d, 3d, . . , mth powers of A. We also write A~ m for (A m )~ l , and if we put A = E we have where m and n are positive or negative integers or zero. If a certain power of the operator A equals the identity, then A is of finite order; and the least positive integer m such that A m = E is called the order of A. We shall prove the following propositions: (a) If A n = E, then n is a multiple of m. For if not, let r be the remainder when n is divided by m, so that n = mq+r, and r<m. Then A n = A m + r = A m A r = E *A r = A r = E, which is contrary to the hypothesis that A m is the least power of A which equals E. (b) The order of A k is m/d, where d is the highest com mon factor of m and k. For, the order of A k is the least positive integer t for which E= (A fc ) t = A kt ; i.e., for which kt is a multiple of m, by (a) . Hence, kt/m = an integer ; and GROUPS OF OPERATORS 33 canceling the common factor d from k and m, the remaining factor of m (namely m/d) must divide t; that is, t = m/d. 27. Finite groups. Generators. We shall now take up the study of sets of operators called finite groups. DEFINITION. A set of g distinct operators Si, S 2 , . . , SginO form a group G of order g if the product of any two of them, whether equal or distinct, is an operator of the set. We write G= (&, S 2 , . . , S g ). If H=(Ti, T%, . . , T g ) is another group containing the same operators as G, we write G = H. We are, how ever, not to infer that the operators are arranged in the same order; that is, G = H does not necessarily imply Si=Ti, . . , Sg=T g . If there is at least one operator in G not found in H , or vice versa, we shall say that the two groups are distinct (G^pH). Example 1. Let O consist of all rotations of a sphere around its center. The three rotations, each of 180, around each of three mutually perpendicular diameters, together with the identity (no rotation), form a group of order 4. Example 2. Let O consist of the different permutations of four points a, b, c, d on a line. The four permutations ( 40): /a b c d\ /ab c d\ /a b c d\ /a b c d\ (abed) \badc) \cdab) \dcba) form a group of order 4. We find, as in 8, that the identity belongs to G; that each operator of G is of finite order; and that the inverse of each operator of G is an operator of G. We shall, obviously, exclude from the concept "group" the (trivial) group consisting of the single operator E. GENERATORS. A set of operators A, B, . . of G having the property that every operator S of G is expres sible as a product of the operators of the set: S= . . A a B b . . A c B d ... , is called a set of generators 34 FINITE COLLINEATION GROUPS of G. Thus, any two of the rotations of 180 in the group of Example 1 above will generate that group. EXERCISES 1. Prove that a set of operators will form a group H if it be known that they belong to a given group and that the product of any two of them, alike or distinct, is an operator of the set H. 2. Show that if A, B, . . belong to a group G, then any product of three or more of these operators, as for instance A~ 1 BA, is again an operator of G. 3. Let Si, Sz, . . , Sg be the different operators of a group G, and let S represent any one of them. Prove that the g operators obtained by taking the products SiS, $2$, . . , S a S are all distinct and are therefore the operators of G over again in some order. 4. Find a set of two generators of the group in Example 2 above. 5. Consider the symbols (xi, yi), . . of Exercise 4, 25, the rule for a product being the first of the two there given. Prove that (0, 1) is a generator of a group of order 4. Prove also that the two symbols (-1, 0), (-1/2, 1/3/2) generate a group of order 6. Both of these groups are abelian (34). 6. Prove that the operators common to two groups form by themselves a group. 28. Subgroups. A group H of order h, all of whose operators are found among thosye of a group G, is called a subgroup of G. Strictly speaking, H is not thought of as a subgroup unless h<g, but we shall here generally under stand the definition in such a way that G is a subgroup of itself (h = g). THEOREM 1. The order of a group G is divisible by the order of a subgroup H of G: g = hk. The quotient g/h = k is called the index of H. The proof is based on our arranging the operators of G in the form of a rectangle of h columns and k rows: E, Sz) i Sh, Vz, SzVz, . . , S h V 2 ; (i) v v k , s 2 v k , , , s h v k . GROUPS OF OPERATORS 35 The first row is composed of the operators of H; Vz is an operator of G not found among those of the first row; Vs is an operator of G not found among those of the first two rows, etc. New rows are added in this manner until every operator of G has been accounted for. It remains for us to prove that the hk operators obtained are all distinct. First, the operators in any one row are distinct. For the assumption S a V c = SbV c gives S a = S b (Exercise 2, 25), which is contrary to the hypothesis that the first row is composed of the distinct operators of H. Second, the operators from two different rows are distinct. For if, say, S a V c = S b V d , c<d, we should have S b - 1 (S a V e )=S b - 1 S b V d ; that is, V d = S V e , where S = Sb~ l S a and therefore belongs to H, since S^ 1 and S a do that. But then V d would occur in a previous row (the cth), contrary to hypothesis. It follows that the table contains all the operators of G, once each. The theorem is therefore proved. We shall indicate the rows symbolically by H, HVz, . . , HVk, and write G = H+HV 2 + . . +HV k . It is to be noticed that the h operators in any one row, except the first, do not form a group. COROLLARY. The order of an operator S of a group G is a factor of the order g. For, if m is the order of S, the m operators E, S, S 2 , . . , >S m ~ 1 form a subgroup of G. Remark. Though the order of a subgroup of G is a factor of the order of G, it is not true in general that there is a subgroup whose order is any given factor of g, unless this factor is a power of a prime number (cf. 36, and Exercise 1, 38). EXERCISES 1. Find the subgroups of order 2 of the group in Example 1, 27. 2. Construct a subgroup of order 2 and one of order 3 of the collineation group given in Exercise 2, 12. 36 FINITE COLLINEATION GROUPS CONJUGATE SETS, 29~31 29. Having defined groups and subgroups, we now introduce the first of two important concepts, conjugate sets and isomorphism. DEFINITION. If A and S are operators belonging to a group, then A and S -1 AS are said to be conjugate. We also say that the first of these operators is transformed into the last by S. The following propositions (a) and (b) are easily verified by the student : (a) There is an operator which transforms S~*AS into A, namely S~*. It follows that the relationship expressed by "conjugate" is reciprocal. (b) If A and B are conjugate, and also B and C, then A and C are conjugate. (Prove that if A is transformed into B by S (B = S~ 1 AS), and B into C by T, then A is transformed into C by ST: C = (ST)- 1 A(ST).) Now let S be any one of the operators of the group G=(E, Szj . . , S g ). Consider the conjugates: (2) E->SE=S, S They are not all distinct: all the operators E, S, . . of G which are commutative with S (and only those) transform S into itself. For, from S = Sa l SS a follows S a S=SS a and vice versa. We therefore select a repre sentative of each distinct conjugate and get what is called a complete set (or simply set) of conjugate oper ators under G. (c) A conjugate set A, B, . . . is transformed into itself by any operator S of G. For, if A = S^SSa, B = Sb l SSb, are distinct conjugates, so are S f ~ l AS t S ~ 1 BS , . . , and if the former set is taken from the series (2), the latter must belong to (2) also, since S - l (S^SS a )S =(S a S )- l S(S a S ), etc. (cf. Exercise 5, 25). GROUPS OF OPERATORS 37 | The number of operators in a conjugate set is deter mined by the following: THEOREM 2. All the operators E, S a , S b , . . of G which are commutative with S form a subgroup H of G, say of order h, and the conjugate set of G to which S belongs contains g/h distinct operators. To prove the first statement, we need merely to show that the operator S a Sb is commutative with S (Exercise 1, 27). This is done as follows: (S a S b )S = S a (S b S) = S a (SS b ) = (S a S)S b = (SS a )S b = S(S a S b ) . To prove the second, consider the table (1), 28. The operators in the first line will all transform S into itself, since they are here commutative with S. All those in the second line will transform S into one and the same (new) operator Vz l SV 2 . For, S a Vz being any operator of HVz, we have Moreover, Vz l SV 2 is distinct from S, as otherwise V 2 would be commutative with S and would therefore belong to H. Similarly, the operators of the third line all trans form S into a third operator V^SVs, distinct from S and from V^SV* Thus, suppose V^SV^ V^SVz; then S would be commutative with V^V^ 1 , so that this operator would be an operator in H, say S a . Hence V s = S a V 2 . But this is impossible, since Fa is not found among the first two lines of the table (1). Proceeding thus, we obtain a distinct conjugate for each line of the table, and the theorem is proved. 30. Conjugate subgroups. If we transform the opera tors E, S 2 , . . , Sk of a subgroup K of G by an operator of G, say V, we obtain the same or another subgroup 38 FINITE COLLINEATION GROUPS which we shall designate V~ 1 KV. For, the resulting k operators belong to G, and the product of any two of them: (V-*S a V)(V-*S b V) = V- l S a VV- l S b V= V~ l (S a S b )V, is contained in the set V~ 1 KV ) since S a Sb is an operator of K. We say that K and V~ 1 KV are conjugate subgroups of G. If the two groups are equal (K= V~ 1 KV; cf. 27), we say that V is commutative with K. The reader may verify the propositions corresponding to (a), (6), and (c), 29, namely that V~ 1 KV is trans formed into K by V~ 1 , that two groups conjugate to a third are conjugate to each other, and that a set of con jugate subgroups of G is transformed into the same set by an operator of G. Finally, the following theorem may be proved in the same manner as the corresponding theorem above: THEOREM 2 . All the operators of G which are com mutative with K (among these are found the operators of K) form a subgroup H of order h , and the number of distinct subgroups of G conjugate to K is g/h . EXERCISES 1. Prove that the operators AB and BA are conjugate. 2. Prove that conjugate operators, or conjugate subgroups, have the same order. 3. If two commutative operators, A and B, are transformed by an operator S, the new operators are also commutative. Hence, the conjugate of an abelian group ( 34) is again an abelian group. 4. If H is the group whose operators are commutative with S (cf. Theorem 2), then T~ 1 HT is the group whose operators are com mutative with T-WT. 5. Prove that two sets of conjugate operators have either no operators in common or are composed of the same operators. Hence show that the operators of G can be separated into distinct conjugate sets, the total number of whose operators is equal to the GROUPS OF OPERATORS 39 order (0) of the group. At least one set contains only one operator, namely the identity. Accordingly, if these sets contain respectively 1, k 2 , fc 3 , . . operators, we have = 1+ &2+fc 3 + . . . Further more, we have (Theorem 2) k 2 *=g/h 2 , k 3 = g/h 3 , . . , so that finally 31. Invariant operators and subgroups. Simple groups. We say that the operator S is invariant under G, or that it is a self -conjugate operator of G, when S is transformed into itself by every operator of G (that is, if the operators (2), 29, are all equal). Similarly, we say that a subgroup H of G is invariant under G or is a self-conjugate subgroup of G, when H = S 2 ~ 1 HS 2 = . . = S g ~ 1 HS g . Thus, the operator S is invariant under H in Theorem 2, and the group K is invariant under H f in Theorem 2 . Any group is an invariant subgroup of itself ( 28). A group which contains no invariant subgroups (except itself) is called a simple group. EXERCISES 1. If G is a simple group, none of the numbers k 2 , /Ca, . . of Exercise 5, 30, can be unity. 2. Prove that a group of order 5 is simple. More generally, a group whose order is a prime number is simple. The least composite number which can be the order of a simple group is 60 ( 48). 3. The operators of a conjugate set of a group G generate an invariant subgroup of G. (Observe that the operators of a group G generated by A, B, C, . . are of the form T= . . . A p B 9 C r . . A^ C* . . . The condition that G be transformed into itself by an operator S of G is that S~ 1 TS belongs to G . Now, if S~ 1 AS, S~ 1 BS, . . are denoted by Ai, Bi, . . , we have (Exercise 3, 25): S~ 1 TS= . . 40 FINITE COLLINEATION GROUPS In other words, if G r is generated by A, B, C, . . , then S-WS is generated by Ai, Bi, Ci, . . . But, these last operators belong to the conjugate set in question if A, B, C, . . do that ( 29), (c) . ) 4. Prove that any two invariant operators of G are commutative, and that their product is also an invariant operator of G. Hence prove that all the invariant operators of G form an abelian group H ( 34) which is invariant under G. Furthermore, any subgroup of H is an invariant subgroup of G. 5. Show that a similarity-transformation belonging to a linear group G is invariant under G, and that the group of similarity- transformations contained in G is an invariant subgroup of G. 6. Prove that the transformations of determinant unity con tained in a linear group G is a self-conjugate subgroup of G. ISOMORPHISM, 32-33 32. It is sufficiently evident from the preceding development that the theory of groups of operators depends entirely upon the scheme according to which the products of operators are tabulated (the "multiplication table" of the group.)* Two groups whose multiplication tables are the same have essentially the same abstract properties, differing only in the notation and possibly the meaning of their operators. The relationship is expressed in the following: DEFINITION. Two groups, G and K, are said to be simply isomorphic when their distinct operators are equal in number and can be arranged in relative order such as : G: E, S 2) $3, . , S g > K: E, T 2 , Tz, . . , T g , so that their products all correspond; that is, if S a Sb = S e , then T a T b = T c . Example 1. The groups of order 4 in Examples 1 and 2, 27, are simply isomorphic. * It will be seen later ( 47) that if a multiplication table is arbitrarily constructed, so that only the conditions of 25 and 27 are complied with, then there is at least one group of operators (a substitution group) whoso multiplication table is the one given. GROUPS OF OPERATORS 41 On the other hand, these groups are not isomorphic with the group G=(E, S 2 , S 3 , 84) whose operations consist in multiplying by 1, i, I, i respectively, where i v\. The square of every operator of the first two groups is the identity; but this is not the case with S 2 or S* of the present group. Example 2. Two conjugate groups, H = (E, A, B, . . ) and V- 1 HV = (E, V~ 1 AV, V-WV, . . ), are simply isomorphic. A more general kind of isomorphism may sometimes be established between two groups, K of order k and G of order g = kh. For instance, let it be possible to arrange their operators in the following manner: K: G: Ti = E; $11, $12, . . , $IA; (3) TZ J $21, $22, , $2A/ Tkt Ski, SfcZ, , Skh, so that first, to each operator of K there correspond h operators of G; and second, to each product $ a a$&/3 = $ CT there corresponds a product T a Tb=T c , irrespective of the subscripts a, /?, y. In such a case we say that G is (h, 1) isomorphic (or multiply isomorphic) with K. Concerning two such groups we have the THEOREM 3. The h operators of G which correspond to the identity of K, namely $n, $12, . . , $IA, form an invariant subgroup of G. First, to prove that $n, $12, . . , Sih form a group H, consider any product $^$15. By the conditions of isomorphism, this product must be an operator in G of the set which corresponds to the operator TiTi = E 2 of K; that is, to E. Hence, there rnust be some subscript c such that $ia$i6 = $ic. Next, to prove that H is invariant under G, we must show that $~ 1 $i a $ belongs to H, where $ is any operator of G. Let $ correspond to T b in K, and $~ 1 $i a $ will 42 FINITE COLLINEATION GROUPS correspond to E, since T b ~ l ET b = E. It follows that S~ l SiaS is to be found in the first line; i.e., it belongs to H. EXERCISE The linear group (6), 7, is (2, 1) isomorphic with the collineation group G=(E,A*) given in the Example, 10. A linear group is always isomorphic with the corresponding collin cation group. 33. Factor group. The group K of order k = g/h discussed in the preceding paragraph, is called a factor group (or quotient group) of the group G of order g, and we write symbolically K = G/H, H being the invariant subgroup of G of order h which corresponds to the identity of K. THEOREM 4. Let G be a group of order g, containing an invariant subgroup H of order h. Then a factor group K = G/H of k = g/h operators can be constructed, to which there fore G is (h, 1) isomorphic. To prove this theorem, we arrange the g operators of G in k lines as shown in table (1), 28, the operators of H forming the first line. Now, the h 2 "products obtained by taking for a pre- factor in turn each of the h operators from a given line (the ath) and as a post-factor in turn each of the h opera tors from the same or another given line (the 6th), will all fall in one line (say the cth; namely the line in which the product V a V b falls); symbolically (HV a )(HV b )=HV c . This is easily seen in the following manner. Let (H) represent the phrase "an operator of H," and we have (H) (H) = (H) ; and, since H is invariant under G, V a (H) = (H) V a . Consequently, (H)V a (H)V b = (H)(H)V a V b =(H)V c , if V a V b = (H)V c . The reader may now readily prove that the symbols H, HV 2 , . . , HVic, looked upon as k distinct operators, GROUPS OF OPERATORS 43 obey the conditions 1 6, 25, as well as the additional condition for a finite group ( 27). For instance, H satisfies the condition for the identity: (H)(HV a ) = (HV a ) (H) = (H) V a . There is therefore at least one group of k operators E, T 2 , . . , T k having the same multiplica tion table as these symbols (47). Hence the existence of the factor group K, as a group of operators, is proved. EXERCISES 1. Let G be a linear group in n variables, and H the group of similarity-transformations contained in G. The collineation group corresponding to G is a factor group G/H. 2. Prove that if the factor group K contains a subgroup K of order A; , then the corresponding hk operators of G form a sub group G of G. 3. If X of the preceding exercise is invariant under K, then G is invariant under G. ^ 4. Prove that the order of an operator of G is divisible by the order of the corresponding operator of K. (Hint: if S belongs to G and S n =E, then the corresponding operator T of K satisfies the equation T n =E.) TWO SPECIAL TYPES OF GROUPS, 34-35 34. An abelian group is a group G whose operators E, Sz, . . , S g are mutually commutative: (4) The following two propositions are immediate con sequences of (4) : (a) Every operator of an abelian group G, as well as every subgroup of G, is invariant under G. (b) A factor group of an abelian group is again an abelian group. Example. Two invariant operators, A and B, of any group G are commutative, since A~ 1 BA=B. Hence, the invariant operators of G form an abelian self -con jugate subgroup of G (Exer cise 4, 31). 44 FINITE COLLINEATION GROUPS THEOREM 5. An dbelian group G of order g contains always a subgroup G whose order g is any given factor of g. In the proof of this theorem we shall adopt the process of complete induction. Accordingly, we assume the theorem true for any group K whose order k is less than the given number g, the order of the given group G for which the theorem is to be proved. Let S be any operator of G. Assuming for the moment that its order h is a factor of g , we construct the group H of order h, consisting of S and its powers, and thereupon the factor group K = G/H. This factor group being abelian and having its order g/h less than g, it contains by assumption a subgroup K of order g /h, a factor of g/h. To K will now correspond a subgroup of G of order g (Exercise 2, 33). / The proof of Theorem 5 thus depends on our finding an operator S in G whose order is a factor of g r . Now, the order n of any operator T chosen at random will either be prime to g , or will contain a factor h which divides g . In the latter case the operator T n/h may be substituted for S of the proof, since it is of order h (26, (6)). In the first case let N be the group formed by T and its powers. The factor group G/N is of order g/n, a number which in the present instance is divisible by g r . This factor group therefore contains a subgroup of order g f , and this again an operator whose order is a factor of g , say h (or g itself). The order of a corresponding operator of G is a multiple of h (Exercise 4, 33), and a power of this operator can therefore be taken for S, as shown above. The abelian groups of order pq are simply isomorphic, p and q representing two distinct prime numbers. To take a special case, let p = 2 and q = 3. An abelian group of order 6 contains an opera tor, A, of order 2 and one, B, of order 3. The operator AB is of order 6 and will generate the group in question. GROUPS OF OPERATORS 45 If p = q the matter is different: there are two distinct types of abelian groups of order p z . One type is generated by an operator of order p 2 ; and the other by two operators, both of order p. If p = 2 the two types are as follows (S is of order 4, A and B both of order 2) : E, S, S 2 , S 3 ; E,A,B, AB. 35. Groups whose orders are powers of a prime num ber. The study of these groups is of the greatest impor tance for the theory of groups in general as well as for linear groups, particularly in view of Theorem 7. We shall here prove the following : THEOREM 6. A group G whose order is p a , p being a prime number, contains p or more invariant operators, forming an invariant subgroup H. If G is not abelian, the abelian subgroup H is contained in a larger abelian subgroup GI which is invariant under G, though the operators of G\ are not all separately invariant under G. 1. To prove the existence of H, we construct all the sets of conjugate operators of G ( 29). If the number of sets that contain just one operator each (invariant operators) is h , and if the remaining sets contain respec tively </ 2 , . . , g n operators, we have (cf. Exercise 5, 30): (5) p = h +gz+ . . . +g n . The numbers gfe, - . , g n are all powers of p, being factors of p a and greater than unity. It follows that h must be divisible by p. Hence, G contains at least p invariant operators, and all such operators form an abelian self- conjugate subgroup (Example, 34), say of order p b (Theorem 1, 28). If G is abelian, p b = p a . 2. To prove the existence of G\, we now construct the factor group K = G/H of order p a ~ b . Treating this in the same way as G above, we find an abelian subgroup 46 FINITE COLLINEATION GROUPS KI consisting of invariant operators of K. Let T be any one of these except the identity, and let p c be the order of that subgroup of K\ formed by T and its powers. This subgroup is invariant under K, and the correspond ing subgroup of G of order p b+c is invariant under G (Exercise 3, 33). We have still to prove that this subgroup Gi is abelian. By referring to the table (3), 32, we see that G\ is gen erated by H and one of those operators, V, of G corre sponding to T of K. Now, all the operators of H are commutative with each other and with V, and the latter is commutative with any power of itself. The operators of Gi are therefore mutually commutative. EXERCISES 1. Prove that a group of order p 2 is abelian, and is therefore isomorphic with one of the types given in 34. 2. The process above may be extended as follows: to K\ corresponds an invariant subgroup H\ of G; the factor group G/H\ contains a subgroup consisting of invariant operators, and to this subgroup will correspond an invariant subgroup # 2 of G, etc. In this manner we shall obtain a series of subgroups of G: H, HI, HZ, HZ, ..,(?. Prove that each is an invariant subgroup of all that follow it, and that the factor groups Hi/H, H^/Hi, . . , are abelian. 3. Prove that a group of order po contains a subgroup of order pt>, if b<a. SYLOW S THEOREM, 36-39 36. In the case of a non-abelian group the sweeping statement of Theorem 5, 34, is not generally true. But we can always predict a certain class of subgroups called Sylow subgroups, as stated in the following: THEOREM 7. (a) Let G be a group of order g, and let p be any prime factor of g. If p n is the highest power of p GROUPS OF OPERATORS 47 that divides g, then there is in G at least one subgroup oj order p a . Denote this subgroup by P. (b) If G contains more than one subgroup P, then all such subgroups are conjugate under G, and their number is of the form l+pk. Proof of (a). We adopt the process of complete induction, assuming that any group whose order is less than the given number g and is divisible by a power of p, say p b , but by no higher power of p, contains a subgroup of order p b . Two cases arise: 1, G contains an invariant operator of order p; 2, G contains no such invariant operator. 1. Let S be an invariant operator of order p, and let H be the subgroup formed by S and its powers. Consider the factor group G/H- Its order is g/p, a multiple of p a ~ l . By assumption, it has a subgroup of order p a-1 . To this subgroup corresponds a subgroup of G of order p a-i p = pa (Exercise 2, 33). 2. Here there may or may not be invariant operators in addition to E in G whose orders are prime to p. All such operators form an abelian group H (Exercise 4, 31). The order, h , of this group is prime to p. For, other wise H would contain a subgroup of order p (Theorem 5, 34), and the arguments of 1 would be valid. We now construct all the complete sets of conjugate operators in G, and obtain an equation corresponding to (5), 35: . +g n . Evidently, the numbers </ 2 , . . , g n cannot all be mul tiples of p, since h is not, whereas g is a multiple of p a . Hence, at least one of the numbers 2 , . . , g n , say 2 , is prime to p. If therefore S is an operator in this set of conjugates, we conclude by the aid of Theorem 2, 29, that G contains a subgroup H of order h = g/g z , a number 48 FINITE COLLINEATION GROUPS which is less than g and is divisible by p a . By assump tion, this group contains a subgroup of order p a which is a subgroup of G, since H is a subgroup of G. 37. In the proof of (6) the following propositions are needed : 1. An operator T of G of order p a which transforms a Sylow subgroup P of G into itself (T~ 1 PT = P) belongs to P. 2. If two different Sylow subgroups, PI and P 2 , have in common a subgroup of order p b , b<a (cf. Exercise 6, 27), then one of them is transformed into just p a ~ b dis tinct groups of order p a by the operators of the other. To prove 1, let ra be the least positive number such that T m belongs to P; in any event, m^p". Then we can show that m is a factor of p a ; for if it is not, let q be the quotient and r the remainder when p a is divided by m, so that p a mq = r; r<m. Then, since T pa and T mq are both contained in P, it follows that T r is also contained in P, contrary to the assumption as to T m . Therefore m is one of the numbers 1, p, p 2 , . . . Assuming that T does not belong to P, let m = p^. The group Q generated by T and the operators of P will now be of order mp a = pP+ a , since it consists of all the operators P, PT t PT 2 , . . , PT m ~\ (That these operators form a group of order mp<* is seen as follows: If we indicate the phrase "an operator of P" by the symbol (P), we have T 8 (P) = (P)T S , since T~ 1 PT = P, and therefore ((P)T 8 )((P)T t ) = (P}(P)T s T t = (P)T s+i , so that the products of operators of the set Q are all contained in the set. Again, the mp a operators are all distinct. For the p a operators PT S are all distinct; and the equation (P)T 3 = (P)T l cannot be true unless (P) = T t ~ 8 ; that is, unless s = l.) But G cannot contain a subgroup Q of order p^+ (Theorem 1, 28). Hence, T must belong to P. GROUPS OF OPERATORS 49 To prove 2, let H be the subgroup of order p b common to the two groups PI and P 2 . By 1, no operator of PI except those of H transform P 2 into itself. We now adopt the process employed in the proof of Theorem 2, 29 : we arrange the operators of PI in the form of the table (1), 28, the first line consisting of the operators of H. To each line of the table there will then correspond a dis tinct group conjugate to P 2 , making in all p a /p b = p a ~ b distinct groups. 38. Proof of (b). Consider a set of conjugate Sylow subgroups of order p a : (6) Pi, P,, . . , P.. By 2, 37, the operators of PI will transform P 2 into p a ~ b distinct groups. If the set contains other groups in addition to these p a ~ b and PI, then the operators of this group (Pi) will transform one of them into say pa-c g rou p s distinct from those already counted. Pro ceeding thus, we find since p a ~ b , p a ~ c , . . are all multiples of p. Now, if G contained another set P n +i, - . , P m of conjugate Sylow subgroups, their number would likewise be of the form l-\-pk . On the other hand, PI is not a member of this new set. We can therefore show that the number of groups in the set is a multiple of p } say pk", in the same manner as the number of groups P 2 , . . , P n were shown to be a multiple of p. But, the equation l+pk = pk" is impossible. It follows that (6) is the only set of Sylow subgroups contained in G, and (b) is fully proved. EXERCISES 1. Prove that a group G whose order is divisible by p b con tains a subgroup of order pb. (Cf. Exercise 3, 35.) 50 FINITE COLLINEATION GROUPS 2. If the largest subgroup H that PI can have in common with another Sylow subgroup is of order p b , prove that the number of Sylow subgroups in G of order pa is of the form 1 -{-p a ~ b k. 3. Show that a group of order 40 contains a single Sylow sub group of order 5, which is therefore an invariant subgroup. 4. Prove that a group of order pq is abelian, if p and q are prime numbers such that p 1 is not divisible by q, nor q 1 divisible by p. 39. We conclude by proving the following useful theorem : THEOREM 8. A subgroup of G of order p b is always contained in at least one Sylow subgroup of order p a . Let K be a subgroup of order p b , and assume it is not contained completely in any one of the groups PI, P 2 , . . , P n , 38. Then, arranging these in sets of conju gates with respect to the operators of K, we find that the number of groups in the set is a multiple of p, as would be the case with the number of groups in the tentative set P n +i) , Pm, 38. But this is impossible by Theorem 7, (6). B. SUBSTITUTION GROUPS 40. Definitions, (a) A substitution (or permutation) S on a given set of letters 0-1, 02, a 3 , . . , a n is the operation of replacing these letters respectively by a p , a q , a r , . . , a w , with the understanding that the sub scripts p, q, r, . . , w are the subscripts 1, 2, 3, . . , n over again in the same or a different order.* We shall temporarily indicate such a substitution by the symbol (7) a q a r * The letters are generally involved in one or more given functions, as a 1 z 1 +o 2 z,+a,z 3 + . . +a n x n . This function would be transformed into o-Zj +a q x 2 +a r x 3 + . . +a w x n by the substitution S above. Tne phrases "replace a by b," "put b in place of a," "change a into 6" are to be regarded as synonymous. SUBSTITUTION GROUPS 51 The columns may evidently be changed about in any manner so long as a p falls below a\, a q below 02, etc. ; and two substitutions are equal if their columns are equal aside from the order in which they are written. For instance, /Oi 02 3\ _ /#2 1 3\ , /! 2 # W a 3 aj ~ \as 2 aj \Oi o 3 ( (b) The product ST of two substitutions /Ol 2 3 0\ T=( ap a 9 a r \a p a q a r . . o./ \a a a& Oy is the substitution resulting from carrying out the two substitutions successively, first S and then T: With this rule for multiplication, the associative law: S(TU) = (ST)U, is readily found to be true. (c) The identity is the substitution in which each letter is replaced by itself: i 02 ai 02 . . o n \ . . aj (d) The inverse of the substitution S is the substitution a r . . a w \ o 3 . . o n / 02 In consequence of these definitions and propositions, the totality of substitutions on n letters ai, a z , . . , a n form a class O of operators (25). 41. Permanent notation. A simpler notation for a substitution S than that employed in (7), 40, may be developed as follows. Writing down the letter a\ we follow with the letter that takes its place, namely a p ; then 52 FINITE COLLINEATION GROUPS we follow with the letter that replaces a p , say a a ; and so on: aia p a a . . . There being a finite number of letters, we must sooner or later arrive at a letter that has already been written. The first one to be thus duplicated must be i/ for, if it were some other letter, say a p , the lower line a p a q . . a w of the substitution S in (7) would plainly have to contain a p at least twice, whereas the letters of this line are the n different letters 0102 . . a n in some order, once each. Let therefore the last letter to be written down before a\ reappears be a m . We then have a cycle (aia p a a . . . a m ). If this cycle does not exhaust the n letters under considera tion, we start with a new letter and proceed as above, forming a new cycle, which may be written immediately after the first; and so on. No one letter will appear in two different cycles, as otherwise such a letter would have to appear at least twice in the lower line of (7). It is cus tomary to exclude all cycles which contain just one letter, such a letter remaining unchanged by S. Thus the sub stitution o _ /01 02 03 04 05 06\ \ai a 3 a 4 02 6 05/ is written S= (020304) (050e). EXERCISES 1. Show that if S-( 0l<wwli ). and T -(******), then \a2030405ai/ VaiasO^s^/ Show also that in the permanent notation, = (0102030405), T = (0203) (o 4 a 6 ), and that ST = (oia 3 a 5 ), TS = (010204). 2. The substitution (020304) (a 5 a 6 ) is the product of the sub stitutions (020304) and (ooOc). Prove that a substitution consisting of a number of cycles is equal to the product of the substitutions SUBSTITUTION GROUPS 53 represented by the individual cycles, and that these factors are commutative. 3. Show that the inverse of the cycle (a\a^ . . a,niQn) is the cycle (a n a n -i . . a^i), and that the inverse of a substitution consisting of a number of cycles is the substitution consisting of the inverses of those cycles. 4. Show that the order of the operator consisting of a single cycle containing n letters is n. Hence prove that the order of a substitution composed of several cycles containing respectively Wi, n 2 , . . letters, is the least common multiple of the numbers Wi, ft-;, . . . (Illustration: the order of (a^scu) (asfle) is 6.) 5. The order of ST in Exercise 1 above is 3, and the cycles in S and T contain respectively 5 and (2, 2) letters. Do these facts agree with the statements in Exercise 4 ? 42. Even and odd substitutions. A cycle of k letters is equal to the product oi kl cycles of 2 letters each (called transpositions) : We therefore classify substitutions into even or odd sub stitutions according to whether they are equal to a prod uct of an even or an odd number of transpositions. For instance, the substitution (aiCfeOs) (a 4 a 5 a 6 a7) is odd, since it can be written as the product (4i<fe)(4itife)(aiai)(aiai) (a 4 a?). This classification is justified since, no matter in which one of the infinite number of possible ways a given sub stitution can be written as a product of transpositions, it is either always even or always odd. To prove this state ment, consider the following function: f=(ai 02) (ai a 3 ) . . (ai ajfe a 3 ) . . (02 a n ) . . (a n _i a n ), * Note that the letter o, is common to all the transpositions. Hence, this is not the normal notation for a substitution as developed in 41. The laws deduced in Exercises 3 and 4 are therefore not applicable to the form above. 54 FINITE COLLINEATION GROUPS namely the product of all the possible differences of the n letters involved. This function changes sign when operated upon by a single transposition, and hence also when operated upon by an odd substitution, but remains unchanged in value when operated upon by an even sub stitution. 43. Substitution groups. Symmetric and alternating groups. A set of g different substitutions form a sub stitution group of order g if the product of any two substitutions of the set, whether equal or not, is again a substitution of the set. Cf. 27, where such a group is given in Example 2. In our present notation this group is written as follows: (8) E, (ab)(cd), (ac)(bd), (ad)(bc). All the n\ permutations on n letters evidently form a group, called the symmetric group on the given letters. Thus, the symmetric group on 3 letters a, 6, c is of order 6 and is composed of the substitutions E, (abc), (ac6), (06), (ac), (be). It is plain that the totality of the even substitutions contained in the symmetric group G on n letters form by themselves a group, called the alternating group. Its order is one-half that of the corresponding symmetric group. For, let there be p even and q odd substitutions in G. The p+q products obtained by multiplying each of these substitutions by a given transposition must pro duce the same p+q substitutions over again (Exercise 3, 27) with this difference, that we now have p odd and q even substitutions. Hence p = q. Inside a given symmetric group, the conjugate of an even substitution is again an even substitution. It follows that the alternating group is invariant ( 31) under SUBSTITUTION GROUPS 55 the corresponding symmetric group. The alternating group on 5 or more letters is always a simple group ( 49). EXERCISE Construct the symmetric and alternating groups on 4 letters a, 6, c, d. Show that the latter contains a single Sylow subgroup of order 4, namely the group (8), and that the former contains 3 Sylow subgroups of order 8, which all contain the group (8). 44. Transitivity and intransitivity. If the letters of a substitution group break up into two or more sets hav ing no letters in common, such that no substitution will replace a letter of one set by a letter of another, then the group is said to be intransitive. Otherwise the group is transitive. The transitive groups are further subdivided into primitive and imprimitive groups. If the letters break up into two or more sets of such a nature that the letters of any one set are either all replaced by letters of the same set, or are all replaced by letters of another set, then the group is imprimitive. If no such division is possible, the group is primitive. Examples. The group (8), 43, is transitive, while the group E, (ab)(cd), is intransitive. The two sets of letters, (a, 6) and (c, d) are called systems (or sets) of intransitivity. The transitive group (8), 43, is imprimitive. Its two systems of imprimitivity may be selected in three ways: 1: (a, 6), (c, d)\ 2: (a, c), (6, d); 3: (a, d), (6,c). The symmetric group on three or more letters is primitive. 45. We proceed to prove the following important theorem concerning transitive groups: THEOREM 9. Let G be a transitive substitution group of order g on n letters a\, 02 . . , a n . Then, (a) there is in G a substitution which replaces any given letter, say a\, by any other given letter; (b) all those substitutions in G which leave unchanged a given letter, say a\, form a subgroup of order g/n. 56 FINITE COLLINEATION GROUPS Proof. (a) Let us assume that ai, 02, . . , a m (m<n) are the letters into which a\ is changed by the various substitutions of G. Then none of these m letters can be replaced by one of the remaining letters a m +i, . . , a n ; or vice versa. For, let V 2 change ai into 0%, and assume that there is a substitution T which changes 02 into Om+i , then the substitution V 2 T would change a\ into Om+if contrary to what was stated in regard to the first m letters. Again, if the substitution T\ changed one of the last n m letters into one of the first m letters, then Ti~ l would do the reverse. But this has just been proved impossible. Now, a transitive group cannot contain two such sets of letters. Accordingly, G must contain a substitu tion (Vz) which replaces ai by 02, one (F 3 ) which replaces ai by as, etc., and finally one (V n ) which replaces a\ by a n . (b) Let all those substitutions which leave ai un changed constitute a set H=(Si, 82, . . , Sh), then H is a group. For, the products S a Sb all belong to H. To find the order of H, we arrange the hn substitutions H, HV 2 , HV 3 , . . , HV n in the form of the table (1), 28. The h substitutions in any one line are evidently all distinct (since S a V c = S b V c would necessitate S a = S b ) , moreover the substitutions of two different lines are dis tinct, since those in the line HV C all replace ai by a c . Finally, any substitution of G must occur in our table. For, if T replaces cti by a c , then TV c ~ l belongs to H, say c - 1 = S ay so that T = S a V c . Hence, g = hn, or h = g/n. C. ON THE REPRESENTATION OF A GROUP OF OPERATORS AS A SUBSTITUTION GROUP 46. Theorem 10. A group of operators G which con tains a conjugate set of n subgroups (or operators) is simply or multiply isomorphic ( 32) with a transitive substitution group on n letters. SUBSTITUTION GROUPS 57 1. If the n subgroups (or operators) are designated ai, 02, . . , a/i, we construct a substitution group K on the n letters i, 02, - . , a n , isomorphic with G, in the following manner. Let S be any given operator of G, and let us suppose that it transforms the group (operator) i into a p , 02 into a q , etc. : Then the subscripts p, q, . . , w are all different and must be the subscripts 1, 2, . , n over again in some order. Accordingly, we can construct a substitution as follows (using the notation of 40) : Ol 02 . . < a p a q . . ( and we shall associate this substitution with S. 2. To the g operators Si, 82, . . of G thus correspond g substitutions Ti, T 2 , . . of a set K; and to prove that these substitutions form a group isomorphic with G it is sufficient to prove that if S a S b = S c , then T a T b = T c (27, 32). Now, if S a transforms a group (operator) a into the group (operator) a", and Sb transforms o" into a ", then S c transforms a into a ": S c ~ l a S c = Again, if T a replaces a by a", and T b replaces a" by a" , then T c =T a T b replaces a by a" . The isomorphism is therefore established. 3. If now the g substitutions obtained above are all distinct, the isomorphism is simple. If, on the other hand, we find that several operators S a i, $02, . . , Sah cor respond to a single substitution T a , then G is multiply isomorphic with the group K composed of the totality of 58 FINITE COLLINEATION GROUPS distinct substitutions in K. For, arranging the operators of G into sets in such a manner that all those operators furnishing the same substitution are thrown into one set, we find that these sets contain the same number of operators. Thus, the h operators SaiS~f , Sa^" 1 , . . , S a hS~* all cor respond to the identity T a T~ a 1 =E; and conversely, if Sn, Sn, . . , Sih all correspond to the identity E, then the operators SaiSu, SaiSu, . . , SaiSih all correspond to one and the same substitution T a Ti = Ta. 4. Finally, to prove that the substitution group thus constructed is transitive, we note that within G the groups (operators) a\ t 0%, a n form a complete con jugate set, so that there is an operator in G which trans forms a\ into any given group (operator) a p . There is therefore within K a substitution which replaces the letter a\ by the letter a p . COROLLARY. // a simple group (31) G contains a set of n conjugate subgroups (or operators), then we can construct a transitive substitution group on n letters which is simply isomorphic with G. EXERCISES 1. The linear group (7), 7, contains a conjugate set of two operators, /? 2 and #4. Construct a substitution group which is (1, 4) isomorphic with the given group. 2. The collineation group of Exercise 2, 12, contains a set of three conjugate operators, namely the 2d, 3d, and 6th. Show that the given collineation group is simply isomorphic with the symmetric group on three letters. 3. Let p, q, r, s denote the four subgroups of order 3 contained in the symmetric group G on four letters a, 6, c, d. Construct the transitive substitution group on the letters p, q, r, s, isomorphic with G. 47. Theorem 11. A group G of order g can be repre sented as a transitive substitution group H on g letters SUBSTITUTION GROUPS 59 (called a regular substitution group). In this representa tion, every substitution except the identity, will replace every letter by a different letter. The group (8), 43, is regular. Let the operators of G, as well as the letters of sub stitution, be denoted by TI (or E), T 2 , . . , T g . We now associate with an operator T p of G the substitution _/Tl T 2 . . Tg\ I rrir rpf rrir ] > \l LI . . lg/ O & where T f a is the letter which in G represents the operator T a T p ; i.e., T a = T a T p . (Cf. 40; the symbol making up the right-hand member of (9) is actually a "substitu tion," since the letters in the lower line are the letters in the upper line written in some order (Exercise 3, 27).) In no case is T a = T a unless T p = the identity =7\; and then every T a = T ai so that Si becomes the identity. Furthermore, no two substitutions corresponding to different operators can be equal (from T a T p = TbT p follows T a = T b ). Hence, if we find that S p S q = S r when ever T p T q = T r , the substitutions Si, . . , S g form a group H, simply isomorphic with G, and fulfilling the conditions of the theorem if it is transitive. Now, since the series T 1} T 2 , . . , T g is equivalent to the series TiT k , T 2 T k , . . , T g T k , aside from the order, the substitution S q may equally well be written O _ I * k ok \ y \ T"* 7^ T^ 7^ 7^ 7^ / V-^l^^^g J-gJ-k^q/ which we shall abbreviate to (T a T k , T a T k T q ). We then have S p S q = (T a , T a T p )(T a T p , T a T p T q ) = (T a , T a T p T q ) = (T a , T a T r )=S r , and the isomorphism is proved. 60 FINITE COLLINEATION GROUPS The group H will be transitive if there is a substitu tion which replaces T\ by any given letter T n . Now, the substitution S n =(T a , T a T n ) does that. The theorem is therefore proved. EXERCISE Construct the regular substitution group on 6 letters x\ t x- 2 , . . , x 6 , isomorphic with the symmetric group on 3 letters. D. ON SIMPLE GROUPS 48. In later chapters it will be of great convenience for us to use a number of known results about simple groups. Some of these results shall merely be stated here without proof; in the case of the theorems 12 and 13 the proofs are outlined for the benefit of advanced students. The detailed analysis would be somewhat lengthy and in part difficult. We begin by enumerating the simple groups whose orders are not greater than 2000* or that can be repre sented as substitution groups on not more than 10 letters :f (a) the alternating groups on 5, 6, . . , 10 letters (49); (6) certain groups of orders 168, 504, 660, 1092. 49. Theorem 12. The alternating substitution group of order n\/2 on n letters is a simple group when ni=5. Outline of proof. 1. If a group on n letters ai, . . , a n contains all the substitutions of the form (a p a q a r ), it is the alternating group. * Holder, Mathematische Annalen, XL (1892), 55; Cole, American Journal of Mathematics, XIV (1892), 378, XV (1893), 303; Burnside, Proceedings of the London Mathematical Society, XXVI (1895), 333; Ling and Miller, American Journal of Mathematics, XXU (1900), 13. Strictly speaking, a group whose order is a prime number is a simple group, having no invariant subgroups. But such groups will not be included under the concept "simple groups" in succeeding chapters. t Jordan, Comptes Rendus, LXXV (1872), 1754. SUBSTITUTION GROUPS 61 2. A possible self-conjugate subgroup of the alternat ing group which contains a substitution S must contain the substitution U = S~ 1 T- 1 ST, where T is any substitu tion in the alternating group. Now, whatever form S may have, it is always possible to find such a substitution T that U is composed of the single cycle (a p a q ar). 3. All the conjugates to U with respect to the sub stitutions of the alternating group must be contained in the self-conjugate subgroup. But these conjugates are indeed all the possible substitutions composed of just a single cycle of three letters each; and therefore, by 1, the pro posed self-conjugate subgroup is the alternating group itself. 50. Theorem 13.* The alternating group on n letters is simply isomorphic with the group generated by the n 2 operators F\, . . , F n - 2 which satisfy the relations: (F a F b y = E (a =1,2, . . , n-4; 6 = a+2, a+3, . . , n-2). Outline of proof. 1. Adopting the process of complete induction, we assume that FI, . . , F n _ 3 generate a group H simply isomorphic with the alternating group on n 1 letters (we take n > 3 ; f or n = 3 the theorem is self- evident). Hence, the following symbols: (10) R n , R n -i, . . , Ri, where R n -i = H, R k (k<n-l) = HF n . 2 F n . 3 . . F k , R n = RiF lt represent at most [(n l)!/2]n = n!/2 operators. * E. H. Moore, Proceedings of the London Mathematical Society, XXVIII, No. 596. The outline given above is of the proof given by L. E . Dickson, Linear Groups (Leipzig, 1901). pp. 289-90. 62 FINITE COLLINEATION GROUPS 2. The symbols (10) contain all the operators generated by ^i, . . , F n _ 2 . For, the set (10) is repro duced if we multiply on the right by any one of these operators, and therefore also by any operator generated by them. Accordingly, the group G generated by FI, . . , F n - Z) is of order nl/2 at most. 3. Now, the substitutions F( = (a^aa) , F^ = (aiCfc) (0304) , ^3=(ai02)(a4a 5 ), > F n-z = (aide) (a n -\a n ) satisfy the re lations imposed upon the corresponding operators of the theorem. It follows that G is isomorphic with the substitution group G generated by F(, . . , Fn-z. But this is the alternating group on n letters, and is of order w!/2. 4. Finally, the order of G being at most w!/2 by 2, it follows that G and G are simply isomorphic. CHAPTER III THE LINEAR GROUPS IN TWO VARIABLES 51. General remarks on linear groups and colline- ation groups. 1. A collineation group may readily be exhibited as a linear group, and vice versa, as shown in 9, 10, 12. It is therefore necessary to discuss only one of these categories, preferably the latter, which lends itself more readily to study. Accordingly, it will be our practice to apply the descriptive terms of chap, ii, (A), to a group under consideration, with reference to this group as written in linear form. For instance, we shall say that the group (7), 7, is non-abelian, though the corresponding collineation group (E, AI, BI, B 2 ) is abelian. Moreover, certain terms have reference to linear groups only, namely (in) transitivity ( 14), (im)primitivity, and mo nomial form ( 60). On the other hand, a group is generally (in the litera ture) described as simple if the corresponding collineation group is simple; that is, when classifying the groups in a given number of variables, we put into the class " simple groups" all the linear groups which are simple, together with those whose factor groups G/H are simple, H repre senting the group of similarity-transformations contained in G. But this is the only exception to the practice agreed upon. 2. It will, furthermore, be our practice to write a linear group in such a way that the transformations all have a determinant unity. Of course, if G (or a subgroup of G) is intransitive, then a constituent of this group embracing one or more sets of intransitivity may not be 63 64 FINITE COLLINEATION GROUPS subject to this rule. An instance is furnished by an abelian group written in canonical form: (a, a" 1 ). Here the groups in one variable, say x = ax f , do not have unity for the value of their determinants. 3. It is often convenient to write the order of a group G in the form g<f> (after Jordan), where g represents the order of the collineation group corresponding to G, and < (not specified) the order of the group of similarity- transformations contained in G. For example, the order of the group (7), 7, may be written either 8 or 4<. We may, correspondingly, write the order of a linear transformation S in the form </<, this being the order of the group generated by S. Thus, 2< is the order of the transformation (i, i), where i = V 1. If the order of the transformation S = (a, /8) is g<j>, that of the trans formation T=(l, /3/a) is g. Obviously we write a linear group in such a way that is as small as possible, under the condition specified in 2. In the case of the groups in two variables, < = 2, except in one instance, since a collineation of order 2 cannot be written in the form of a linear transformation of determinant unity unless it has the form (i, i), where i = V 1. A group of even order must therefore contain the group of similarity-transformations E = (l, 1), E\ = ( 1, 1). The exception mentioned is an abelian group of odd order. 4. Equivalence. A group is said to be equivalent to all the groups which flow from it by means of a change of variables. If the groups K\ and K 2 are equivalent, then there is a linear transformation T such that T~ l KiT = K 2 ( 13). Two groups equivalent to a third are equivalent to each other (cf. 30). 52. The linear groups in two variables: mode of attack. There are several processes available for the determination of the different non-equivalent groups in LINEAR GROUPS IN TWO VARIABLES 65 two variables.* We shall here employ a modified form of Klein s original process, for the sake of its historical and geometrical interest. An outline of this process follows. 1. Any given transformation S of a group G in the variables x\, X2, whose Hermitian invariant is I = XiXi-{- X2X2 can be written as a product S = 818283,^ where Si and $ 3 have the canonical form: 81 = (cos ui sin u } cos u-\-i sin u), S3=(cosw i sin w, cos w+i sin w), and Q _ [ cos v sm v ~\ L sin v cos v J * The transformation S of the group G conjugate-imaginary to G is similarly equal to the product 818283, where 81= (cos u+i sin u, cos ui sin u), $ 3 = (cos w-\-i sin w, cos w i sin w?), (53) We shall denote by_H the intransitive group whose constituents are G and G; that is, the group whose vari ables are Xi, fy, x\ y x 2 . In these variables let T, TI, T 2 , T 3 denote the transformations corresponding to S, Si } 82, 83, and we have T=TiT 2 T 3 . 2. Now let the three real functions X = XiXi 2^2, Y = XiXz + xixi, Z = (xiXz 2X1) V 1 denote rectangular co-ordinates in space. The transformations 7\, T 2 , T 3 * Klein, Mathematische Annalen, IX (1876), 183 flf.; Vorlesungen ilber das Ikosaeder, Leipzig, 1884, pp. 116-20; Gordan, Mathematische Annalen, XII (1877), 23 ft.; Jordan, Journal fiir die reine und angewandte Mathe- matik, LXXXIV (1878), 93-112; Atti delta Reale Academia di Napoli, VIII (1879); Puchs, Journal fiir die reine, etc., LXXXI, LXXXV (1876, 1878), 97, 1 flf.; Valentiner, De endelige Transformations-Cruppers Theori, Copenhagen, 1889, pp. 100 ft*. t The transformations S lt S. 2 , S 3 are not separately transformations of G. 66 FINITE COLLINEATION GROUPS are then shown to represent real rotations around the X , Z , X axes respectively. It follows by a well- known theorem that T is a real rotation around a certain axis which passes through the origin. There results a group G of rotations in space which is isomorphic with G ( 54). 3. Since the rotations of G leave the origin fixed, they must transform into itself a sphere 2 whose center is the origin. If now R be an axis of rotation and if PI be one of the points where R cuts 2, then PI, and all the points Pz, . . , P t into which PI is transformed by the rotations of G will be the vertices of a regular poly hedron (including the limiting cases where there is a single axis of rotation or where the polyhedron becomes a flat polygon) (55). 4. The groups of rotations (G ) may now be con structed to correspond to the regular polyhedra. We find five types, and correspondingly five non-equivalent linear groups (G) ( 56-58). 53. Proof of 1. The transformation S has the unitary form and its determinant is unity. We may therefore write (cf. Exercise 1, 20): _T a 6] 1-6 aj aa-\-bb=l. Let p and q represent the positive square roots of the real positive numbers aa and bb respectively, and we have p 2 -\-q z =l. Accordingly, we can write p = cos v, q = sin v. Moreover, \a/p\ = 1 and \b/q\ = 1, so that we may write a/p = cos h i sin h, b/q = cos k i sin k; a/p = cos h+i sin h, b/q = cos k+i sin k. If we finally put h = u+w t k = u w, we obtain by direct multiplication = S. LINEAR GROUPS IN TWO VARIABLES 67 54. Proof of 2. The functions X, Y, Z will be transformed by 7\, T 2 , T 3 , defined in 1, into linear func tions of themselves. In fact, we may readily prove that (Z)Ti= Y sin 2u+Z cos 2w, with similar results for T 2 and TV We can exhibit these results in a different form. Looking upon TI, Tz, T 3 as linear transformations in the variables X, Y, Z, they will appear as follows : TV X = X , Y=Y f cos 2u-Z sin 2w, Z=Y sin 2u+Z cos 2w; TV X = X cos2v+Y sm2v, Y= -X sin 2v+F cos 2v, T 3 : X = X , Y=Y cos 2w-Z sin 2ti>, Z=Y f sm2w+Z f cos2w. If we interpret X, Y, Z as rectangular co-ordinates in ordinary space, we recognize here three real rotations around the X, Z , X axes respectively. These rotations, performed successively, are equivalent to a single rotation T. To the different transformations (S) of G will in this manner correspond rotations (T 7 ) of a group G , isomorphic with G. The isomorphism is (1, 2) in case G contains the transformation Ei=(l, 1), since both this transforma tion and the identity (E=(l, 1)), and no others, give rise to the single rotation E =(l, 1, 1), and vice versa. 55. Proof of 5. Consider the sphere 2 together with all of the axes of rotations of G . A rotation B of G will permute these axes among themselves; in fact, the axis of a rotation A is by B transformed into the axis of the rotation B~ 1 AB. Accordingly, if PI is the extremity 68 FINITE COLLINEATION GROUPS of an axis of period m (that is, an axis whose corre sponding angles of rotation are the different multiples of 360/w), then the points PI, P 2 , . . , Pt into which PI is transformed by the various rotations of G will be extremities of axes of period m, and the distribution of these points about any one of them is similar to the distri bution about any other. Now, if t> 1, let arcs of great circles be drawn connect ing PI with all the points P 2 , . . , P*, and let the shortest arc be of length L. The number of arcs of this length radiating from PI is m or a multiple of m, since always m of the arcs are interchanged by rotations about the axis through PI. However, there cannot be more than 5 such arcs, unless L = 180. For, if there were 6 or more, a pair of them, say C s , C r , would make an angle 0=i60 with each other at PI; and, this being the case, the arc L connecting P s and P r (the points of P 2 , . . , P t located on C a and C r ) would have a length <L. For, by trigonometry, cos L = cos 2 L+sin 2 L cos fli^cos 2 L-\-\ sin 2 L = | (l-f-cos 2 L)>cosL; and, since 0<Z/<90, it follows that L <L. But this is contrary to hypotheses, since the lengths of the arcs ra diating from P 8 are equal to the lengths of the arcs radiating from PI. Similarly we may prove that an arc of length L joining two of the points PI . . P* cannot intersect another arc of the same nature. Let m>2. Then it follows that there are just m arcs of length L radiating from PI, each making an angle of 360/ra with its adjacent arcs. The same is true for each of the points P 2 , . . , Pt, and we see that the sphere will be divided by all the arcs of length L, joining the various points PI . . P t which can be reached from one of them by passing along such arcs, into a number of LINEAR GROUPS IN TWO VARIABLES 69 equal and regular polygons. Accordingly, these points are the vertices of a regular polyhedron inscribed in 2. The diameters of S passing through the middle points of the arcs L and through the middle points of the regular polygons will either coincide with the axes already obtained or will be additional axes of rotations of (? . Next, let there be no axis of period greater than 2. If there are two axes of period 2, say AI and A 2 , cutting each other under an angle a which may be assumed ^90, then a rotation of 180 around AI followed by a rotation of 180 around A 2 is equivalent to a rotation of 2a around an axis perpendicular to the plane of AI and A 2 . Hence 2a is a multiple of 180; i.e., a = 90. It follows that we have just one axis of period 2, or just three such which are mutually perpendicular. THE GROUPS OF THE REGULAR POLYHEDRA, 56-58 56 (4) . Limiting cases. In order to tabulate the linear groups (G) we may proceed as follows. From the geo metrical data given the analytical equivalents of the rota tions of G can be calculated, and then we can reverse the processes of 1 and 2. This work may be facilitated by placing any given configuration arrived at in 3 in any convenient position with reference to the axes of co ordinates X, Y, Z, since such a shifting of the figure is equivalent to a change of variables (x\, 2) in the respective group G. Beginning then with the simplest case where there is a single axis of rotation, we let this be the X axis. Then sin 2i> = and cos2y=l. Hence S has the canonical form (a, a); aa 1. (A) G : a single axis of period g; G: an abelian group (intransitive) of order g: S m =(* m ,a- m ); m=l,2, . . ,g; a = l. 70 FINITE COLLINEATION GROUPS We next have a group G containing the axis of period g in (A), in addition to g axes of period 2 lying in the plane X = 0. Let one of the latter be the Z axis ; we here have cos2z;= 1, cos 2(u w) = l, and the corresponding transformation of G is found to be w= (B) Dihedral group. G : one axis of period g and g axes of period 2 ; G: an imprimitive group of order 20< consisting of the transformations m-1, 2, . . , g; a* = l. 57. The tetrahedron and octahedron. We now examine the five ordinary regular solids. Of these, the hexahedron and octahedron furnish the same set of axes of rotation, as do also the dodecahedron and icosahedron. We therefore have only three cases to consider : the tetra hedron, octahedron, and icosahedron. In the case of the tetrahedron we have four vertices and correspondingly four axes of rotation of period 3; besides, three axes of period 2, each passing through the middle points of a pair of opposite edges. The latter are mutually perpendicular and may be taken as the X, Y , and Z axes. The corresponding transformations of G are then as follows: either directly or after multiplication by E\=( 1, 1). If the vertices are named a, b, c, d, the three rotations LINEAR GROUPS IN TWO VARIABLES 71 permute them among themselves according to the sub stitutions (ab)(cd), (ad)(bc), (ac)(bd). The remaining rotations permute the vertices three at a time cyclically, as (abc), . . . The corresponding transformations of G may be determined analytically from the conditions that they are each of order 3 and transform the collineations corresponding to Wi, Wz, Wz cyclically. Certain ambiguities arise from the fact that the similarity- transformation EI = ( 1, 1) is present in the group. Thus, S = (abc) may transform W\ into Wz or into WzEi, etc. There results four possible forms for S, all of which are present in G if one of them is. We shall choose the following form : -l+i -l+i 2 2 l+i -l-i 2 2 (C) Tetrahedral group. G : generated by (abc) and (ab)(cd); G: a group (primitive; 60) of order 120 generated by the transformations Wi and S above. The rotations of the octahedron include those of the tetrahedron (abc) and (ab)(cd) if here a, b, c, d represent each a pair of opposite faces. To the list of generating rotations we now add one, U say, having the same axis as Wi, but being of period 4: (acbd), or U 2 =Wi. The corresponding transformation is easily determined from this last equation. We find that it has the canonical form U v/2 1/2 A 72 FINITE COLLINEATION GROUPS (D) Octahedral group. G : generated by (abc) and (acbd) ; G: a group (primitive) of order 24<, generated by S and U above. 58. The icosahedron. Counting the rotations of an icosahedron we find 1 axis of period 1 (the identity), 15 axes of period 2, 20 axes of period 3, and 24 axes of period 5, making 60 in all, the order of G . A Sylow subgroup of order 4 ( 36) must be repre sented by three mutually perpendicular axes of period 2 (55); moreover, two distinct subgroups of order 4 can have no. axis of period 2 in common, since otherwise such an axis would be of higher period. Hence, the 15 axes of period 2 must belong to 5 subgroups of order 4. It is readily observed that no rotation of G can trans form each of these subgroups into itself. It follows that G can be written as a substitution group on 5 letters a, 6, c, d, e, simply isomorphic with G f (46). The 20 rotations of period 3 are represented by all the cycles on three letters; that is, the substitution group in question is the alternating group on 5 letters ( 49). Now, this group is generated by the following operators (50): Fi, F z , F 3 (corresponding to the substitutions (abc), (ab)(cd), (ab)(de)), which satisfy the relations For the corresponding transformations of G, we may evidently take respectively S, W\ above, and a new transformation V which fulfils the conditions = E or Ei V 2 = or LINEAR GROUPS IN TWO VARIABLES 73 The first and last ambiguities fall away, since of neces sity V*=(SV)* = Ei (cf. 51, 3); and by using VEitt necessary instead of V we may take (WiV) 3 = E. We then find v= ~2 where l-J/5 = 1 + 1/5 (E) Icosahedral group. G : generated by certain rotations of periods 3, 2, 2, corresponding to the substitutions (o6c), (ab)(cd), (ab)(de) , G: a group (primitive) of order 60< generated by S, Wi, and V above. The group (E) may be given the following useful form. Cor responding to the substitutions S = (abcde), t/ = (od)(6c), T = (ab)(cd) of the alternating group on five letters, a set of generators are constructed from the relations S 5 = E, U - 1 S U = S - l t T 2 =Ei, U - 1 T U = T or =T Ei. The transformation S f is at the outset written in canonical form ; during the subsequent determinations of U and T we simplify undetermined coefficients as much as possible by suitable changes of variables. The two types of groups (E) here given are equivalent (51). 59. Jordan s process.* We shall in conclusion give an outline of Jordan s method for determining the linear groups in two variables. * Jordan, op. cit., in footnote to 52; Valontinor, op. cit., in the same footnote. 74 FINITE COLLINEATION GROUPS 1. It can be proved without difficulty that if two different abelian groups in two variables have in common a transformation S which is neither E nor E\, then the transformations of the two groups are mutually commuta tive, so that they both belong to a single abelian group. (Writing S in canonical form, we find that a transforma tion which is commutative with it has the canonical form also.) 2. Let KI be an abelian group written in canonical form. Then if T transforms KI into itself (T~ l KiT = K 1 ; cf. 30) and is not commutative with each of the trans formations of Ki t we can readily prove that it must inter change the variables of K\ (i.e., T has the form , 3. Now let G be a linear group in two variables and of order g<j>. By 1, we can sort its transformations into distinct abelian subgroups K\, K z , . . , K m , of orders &i<, kz<f t . . , fc m <, such that no transformation, except E and EI, occurs in two distinct subgroups. Hence we have 0* = <H-(fci- l)* + (k- !)<#>+ - - +(fc w -l)* ; (It is observed that E and EI are counted once each in the term < in the right-hand member, but not in any of the numbers (ki !)<, . . .) 4. The various groups K\, K 2 , . . , K m may be distributed into conjugate sets. Thus, if KI, K 2 , . . , K^ make up one such set, the orders ki<f>, &2<, . . are equal, and the subgroup H which contains K\ invariantively (cf. 30) is, by 2, of order 2&i< or ki<j> } according as there is, or is not, a transformation of type T in G interchanging the variables of KI. Hence, if the subgroups are arranged in conjugate sets as suggested above, we get LINEAR GROUPS IN TWO VARIABLES 75 which reduces to m i 2k" This diophantine equation is now shown to have a finite number of solutions for the integers g, k , k", . . , and by the aid of these solutions the various groups G may be determined without much trouble. EXERCISES 1. Construct the linear transformations in the variables X, Y, Z (cf. 54) corresponding to the generating rotations Wi, W 2 , W 3 , S, U and F, 57-58. Hence show that the groups (C) and (D) have the monomial form ( 60) when written in the variables X, Y, Z. 2. Verify the diophantine equation (1) for the groups (A) to (E). (Hint: to each abelian subgroup of G corresponds a single axis of rotation of (/ .) CHAPTER IV ADVANCED THEORY OF LINEAR GROUPS* A. ON IMPRIMITIVE GROUPS AND SYLOW SUBGROUPS 60. Primitive and imprimitive groups. Let us sup pose that a group G contains not only transformations of the type given in the example in 14, but also some of type p q t v q p v t u w r s w u s r which upon the change of variables employed in 14 becomes p-q t-v u w rs p+q t+v u+w r+s then we say that G is imprimitive, under the assumption that it is transitive. In general, a transitive group G, in which the variables (either directly or after a suitable choice of new variables) can be separated into two or more sets Y\, . . , Y k , such that the variables of each set are transformed into linear functions of the variables of the same set or into linear functions of the variables of a different set, is said to be imprimitive. If such a division is not possible, the group * Wo observe the rules laid down in 51, with the exception that the transformations discussed in 60-62 are not restricted to be of determi nant unity. 76 ADVANCED THEORY OF LINEAR GROUPS 77 is primitive. The sets FI, . . , F* are called sets of imprimitivity . THEOREM 1. The n variables of an imprimitive linear group G may be so chosen that they break up into a certain number of sets of imprimitivity YI, F 2 , . . , F& of m variables each (n = km), permuted among themselves accord ing to a transitive substitution group ( 44) K on the k letters FI, . . , Y k , isomorphic with G. To the subgroup of K which leaves the letter YI unaltered ( 45) there cor responds a subgroup of G which is primitive as far as the variables of the set YI are concerned. If m=l, k = n, then G is said to have the monomial form or to be a monomial group. Proof. The transformations Si, &, . . , Sh in G which transform the variables of each set into linear functions of the variables of that same set (there is at least one such transformation, namely E), form a group; moreover, if V is any transformation in G which permutes the sets in a certain way, then SiV, . . , ShV, and no others, permute them in the same way. If we therefore associate with each such system of h transformations a substitution on the letters FI, . . , F*, indicating the manner in which V permutes the corresponding sets of imprimitivity, we obtain a substitution group K, to which G is (h, 1) isomorphic ( 32). Since G is a transitive linear group ( 14), K must be a transitive substitution group ( 44). Hence, K con tains k 1 substitutions T 2) T 3 , . . , T^ which replace FI by F 2 , F 3 , . . , Ffc respectively ( 45). Let us select kl corresponding transformations of G and denote them by 2, F 3 , . . , Vk. The conditions that their determinants must not vanish is found to imply that the sets contain an equal number of variables, say ra, so that n = km. (A simple example will suffice to show this clearly, say n = 4, k = 2.) 78 FINITE COLLINEATION GROUPS There is in K a subgroup KI whose substitutions leave YI unaltered (45). This subgroup, together with the kl substitutions corresponding to V%, . . , Vk will generate K. Correspondingly, G is generated by Vz, . . , Vk and that subgroup G\ whose transformations replace the m variables of the set YI by linear functions of the same variables, however they may permute the remaining sets. Let us fix our attention upon just that portion of each of the transformations of G\ which affect only these m variables, and which may be looked upon as forming the transformations of a linear group [(jj. We shall proceed to prove that if [G\] is not primitive, then the variables in G may be changed in such a way that the number of new sets of imprimitivity is greater than k. Assuming then that [Gi] is not primitive, its m vari ables are found to break up into at least two subsets of intransitivity or imprimitivity (either directly or after a change of variables) . For the sake of definiteness we shall assume just two such sets, Y{, Y". New variables will now be introduced within the sets F 2 , . . , Yk such that V 2 will replace Y[, T{ by two distinct sets Y z , Y i in F 2 ; and so on for each of the transformations V 3 , V*, . . , V k . Let us temporarily write (Y r )TY 8 to represent the phrase " T transforms the variables of the set Y r into linear functions of the variables of the set Y 8 "; and (Y r , Y J)T=(Y S , F /) to represent the phrase "T trans forms the variables of any subset of Y r into linear func tions of the variables of a subset of F 8 " (that is, either (Y r )T=Y ., (r r )!T=F /; or (F;)T=F /, (Y^T=Y 8 ). Then the properties of the 2k subsets relative to the transformations F 2 , . . , V k may be stated in the form : (FJ, FY)F=(F{, r/), or its equivalent form: (FJ, F /) Fi 1 = (Y{, F /); = 2, . . , k. It remains for us to prove that any transformation W in G will permute these subsets among themselves; that is, ADVANCED THEORY OF LINEAR GROUPS 79 assuming (F p )Tf=F g , then we have also (Y p , Y P )W = (Yq, Yq). To show this, consider the transformation This transformation belongs to GI, since it transforms the set YI into itself: Hence, by assumption, (Y[, Y {)T=(Y[, F /); and, since V p W=TV q , so that (71, Y {)V p W=(Yi, Y {)TV q , we derive (Y p , Y p )W=(Yi, Y q ), what we set out to prove. Accordingly, if [Gi] is not primitive, we can change the n variables of G so as to increase the number of sets. This process may be continued until the sets contain just one variable each, or until we get a group [Gi] which is primitive. Hence the theorem. EXERCISES 1. Prove that there is only one type of imprimitive groups in three variables, namely the monomial type. Prove also that there is a single non-monomial type of imprimitive groups in four variables. 2. Prove that the group generated by S = ( 1, 1, 1) and T: x l =-i(x[+2xi+xi)/2, Xt = (-x[+xi)/2, x t = i(x[-2xL+xb/2, is imprimitive. (Hint: By Exercise 1, the group must be monomial. Hence, if the new variables are x, y, z, the function xyz is a relative invariant ( 88) of both S and T.) 61. Sylow subgroups. LEMMA. A linear group G which contains an invariant abelian subgroup H not composed entirely of similarity- transformations, is either intransitive or imprimitive. Proof. Write H in canonical form. The variables can then be arranged in sets Xi, X z , . . . , having the property that a transformation of H affects all the vari ables of any one set by the same constant factor. To illustrate, let H be generated by the transformations i = ai, O!, a lt O!, aj r, = (j8,, ft, ft, ft, ft) (ft* A). Here we have three sets: Xi = (x lt x 2 ), X 2 = (x 3 , z 4 ), X 3 = (x 6 ). 80 FINITE COLLINEATION GROUPS We shall for the moment write the phrase " T trans forms the variables of the set X p into one and the same constant multiple of themselves" symbolically: (X P )T = c(X p ). Thus, in the illustration given, (X 2 )T 2 = c(X 2 ), the constant multiple being /3 2 . More generally, if X denotes an aggregate of certain linear functions y iy y 2 , . . of the variables of the group, the equation (X )T = c(X ) implies that (yi)T = ayi, (y 2 )T = ay 2 , . . , a being a constant, the same for every function y\, y 2) . . . Then if (X )T = c(X r ) for every transformation T of H, it follows that X cannot contain variables from two or more distinct sets X\j X 2) . . . For, otherwise at least one transforma tion in H would affect some of the letters involved by one constant factor, and others by a different constant factor. Now let S and T be any transformations from G and H respectively. The variables of a given set X p are by S transformed into linear functions of the variables of G, forming an aggregate which we shall denote by X . Since H is invariant under G, the transformation V = STS~ l belongs to H, and we have (X p ) V = c(X p ) . Hence, (X P )ST=(X P )VS, or (X )T = c(X f ). Accordingly, the linear functions of X r must contain variables from only one set of H; in other words, S trans forms the variables involved in X p into linear functions of the variables of some one set of H. The sets X\, X 2) . . are therefore permuted among themselves by S, and the lemma is established. THEOREM 2. A linear group whose order is the power of a prime number can be written as a monomial group by a suitable choice of variables Xi, x 2 , . . , x n ; that is, its transformations have the form:* x s a st x t , * First given by the author in Transactions of the American Mathe matical Society, V (1904), 313-14; VI (1905), 232; see also Burnside, Theory of Groups of Finite Order. 2d ed., Cambridge, 1911, p. 352. ADVANCED THEORY OF LINEAR GROUPS 81 where s and t run through the numbers 1, 2, . . , n, though not necessarily in the same order. Proof. 1. A group P whose order is the power of a prime number p is either abelian or it contains an invariant abelian subgroup Q whose transformations are not sepa rately invariant under P ( 35). In the first case the the orem follows from Theorem 10, 22. In the second case, Q can be written in canonical form, and P is intransitive or imprimitive by the above lemma. If the theorem is true for a transitive group, it is evidently true for an intransitive group; hence we need merely consider the case where P is imprimitive. 2. By Theorem 1, 60, P is monomial unless there is a group [Pi] which is primitive in the m variables of a set YI. But the latter alternative is impossible by 1, since the order of [PJ is again a power of p, being a factor of such a number. Hence the theorem. COROLLARY. A linear group P in n variables, whose order is a power of a prime number p which is greater than n, is abelian. Proof. Write P in monomial form. Then any trans formation T which does not have the canonical form will permute the variables Xi, x%, . . , x n in a certain manner, indicated by a substitution S on these letters. If the order of S is k, that of T is k or a multiple of k. Now, the order of T is a power of p ( 28, Corollary) ; it follows that k is a power of p. But this is impossible since k is n or a product of numbers all less than n (cf . Exercise 4, 41), and none of the prime factors involved can be p. Hence, every transformation of P has the canonical form, and this group is accordingly abelian. EXERCISE If pb is the highest power of p which divides n\, prove that a Sylow subgroup of order p a in n variables contains an invariant abelian subgroup of order pa-b a t least, if a>b. 82 FINITE COLLINEATION GROUPS 62. Theorem 3. A linear group in n variables and of order g = g p a q b r c . . , where p, q, r, . . are different primes all greater than n+1, contains an abelian subgroup of order p a q b r c . . . To prove this theorem by complete induction, we assume it true for any group in less than n variables and for any group in n variables whose order is less than g. That the theorem is then true for an intransitive group in n variables will be shown below (1). There is left the case of a transitive group G of order g in n variables. It may be assumed that the determinants of the transforma tions of G are all unity (cf. 2 below). At the outset we anticipate a theorem given later (cf. Exercise 2, 90). From this it follows that G con tains an abelian subgroup HI of order pq. This subgroup contains a transformation S of order p which is also contained in a Sylow subgroup P of G of order p a ( 39). The groups HI and P being abelian, S is invariant under both, and will therefore also be invariant under the group K generated by Hi and P, whose order is divisible by p a q. Now, S cannot be a similarity-transformation (by 2; otherwise its determinant could not be unity) ; it follows that K is intransitive (cf. proof of Theorem 10, 22), and must therefore, by 1, contain an abelian subgroup H 2 of order p a q. Again, this subgroup Hz and a certain Sylow subgroup Q of G of order q b have in common a transformation T of order q, which must be invariant under both. We find, by the process above, an abdian subgroup H of order p a q b . In the same way we obtain abelian subgroups H" } . . of orders pr c , . . . Now, the subgroups of order p a form a single conjugate set within G. We may therefore select such conjugates of H", H ", . . , that these conjugates have in common with H the group P. The group generated by them will be intransitive, the operators of P being invariant, and ADVANCED THEORY OF LINEAR GROUPS 83 its order will be a multiple of p a q b r c . . . The theorem follows by 1. 1. Consider an intransitive group G in two sets of intransi- tivity, the corresponding component groups being designated G and G". In G we find, by assumption, an abclian subgroup K of order k = p a qPr y . . , which we shall write in canonical form. Let the identity in G correspond to a group G { in G", of order g"; the order of G will be the product of the orders of G and G" (32). Now, in G i we find an abelian subgroup K" of order p^qT* . . ; writing this in canonical form we see that the group generated by K and K" is the group sought. * 2. If the determinants of the transformations of G are not all unity, we construct a group G\ by the method of 12 whose transformations have this property. If then we find in Gi an abelian group of order p a qPr y . . , we evidently have a correspond ing group in G, also abelian. The latter may contain similarity- transformations whose orders are prime to paqb r c . . but these transformations can easily be disposed of ( 34). EXERCISES 1. If n-f lisa prime, andif theorderof Gisg = g (n-{-l)tpaqb . . t where t>l, and p, q, . . are primes greater than n + 1, prove that there is in G an abelian subgroup of order (n-\-\yp<*q*> . . . 2. It follows from Theorem 3 that a group in n variables whose order is not divisible by a prime factor smaller than n+2 is abelian. Prove that if the order is not divisible by a prime factor smaller than n+1, the group is abelian. B. ON THE ORDER OF PRIMITIVE GROUPS 63. The remainder of this chapter will be devoted to the discovery of limits to the magnitude of the prime factors that may enter the orders of primitive groups ( 63-64), to the highest admissible powers of the prime factors under certain conditions ( 65-68), and to the orders of abelian subgroups in general ( 69-73). The chapter closes with a discussion of the order of a primitive group (74). THEOREM 4. No prime number p>7 can divide the order of a primitive group G in three variables. 84 FINITE COLLINEATION GROUPS The method of proof consists in showing that if the order g contains a prime factor p>7, then G is not primi tive. We subdivide this process into four parts as fol lows: 1 proving the existence of an equation F = Q, where F is a certain sum of roots of unity; 2 giving a method for transforming such an equation into a con gruence (mod p) ; 3 applying this method to the equation F = 0; 4 deriving an abelian self-conjugate subgroup P of order p k . 1. The order g being divisible by p, G contains a Sylow subgroup of order p a and therefore a transformation S of order p. We choose such variables that S has the canonical form Two cases arise : two of the multipliers are equal, say o t = 03, or they are all distinct. They cannot all be equal, since a\ = 1 and a? = 1 imply a : = 1 ; whereas S is not the identity. Of the two cases we shall treat the latter only; the method would be the same in the former case,* and the result as stated in Theorem 4 would be the same. Selecting from G any transformation V of order p: [ai bi Cil 02 6 2 c 2 , as &s c 3 J we form the products VS, VS 2 , VS*. Their character istics ( 23) and that of V will be denoted by [VS], . . . , [V], and we hive [V] =ai [VS] =a* * The congruence (7) would hero be of the first degree in /*. ADVANCED THEORY OF LINEAR GROUPS 85 We now eliminate ai, 62, c 3 , from these equations, obtaining [V] 111 [VS\ Oil 0.2 "3 (2) a? a 2 = 0. Expansion and division by (a! a 2 )(a 2 a 3 )(a 3 a x ) gives us (3) [V&]+K[V]+L[VS\+M[V&] = Q, K, L, M being certain polynomials in a b a 2 , a 3 ( 132), with the general term of type afa^af. Since a l} a 2 , a 3 are powers of a primitive pth root of unity a ( 133), the quantities K, . . . are certain sums of powers of a. Moreover, the characteristics [V], . . . are each the sum of three roots of unity (Exercise 8, 6). If, there fore, the products in (3) were multiplied out, there would result an equation of the kind discussed in 133, 6. The various terms could therefore be rearranged in sets as explained in that paragraph, which gives us an equation of the form A, B, C, . . . being certain sums of roots of unity; a, /?, y, . . . primitive roots of the equations x p =l, x q =l, x r =l, . . . respectively; and p, q, r, ... different prime numbers. The coefficients A, B, C, . . . may be put into certain standard forms. Thus, any root c4=l occurring in any of these sums will be assumed to be resolved into factors of prime-power indices ( 133, 4): c = c p c y r . . , the root p being of index p m , c g of index q n , etc. Further more, within A any root c p will be assumed to be either unity or a root whose index is divisible by p 2 . For, if 86 FINITE COLLINEATION GROUPS t p were of index p, say p = a fc , we could put 1 in its place, since a 2_|_ ^ f ^ _f_ a p-l)= a *_J_ a *+l+ a *+2_|_ ^ ^ ^ by means of the relation a? = 1. Likewise we assume that any root e g within J5 is either equal to unity or is a root whose index is divisible by q 2 ; and so on. To illustrate, take p = 2, <? = 3, and let i be a root of index 4 (12= _ i) an d r a root of index 9 (r 3 = w, w 3 = l). Then the standard form for the expression (5) (-l a;-l)(l would be 2. We shall now make certain changes in the values of the roots in the equation (4). First we put for every root g , CT, . . . whose index is divisible by the square of a prime other than p 2 (as r 5 in the example above), leaving undisturbed the roots whose indices are not divis ible by such a square, as a, a 2 , . . . , /?, . . . . Al though these changes will turn the quantities A, B, . . . into certain new sums A , B f , . . . . , the equation (4) is still true, since the vanishing factors l-f-a-f-a 2 -|- . . . +aP~ 1 , etc., have not been affected. Next we put in place of q 2 of the roots ft /3 2 , . . . , fi q ~ l , and 1 for the remaining root, thus chan ging B (l+P+!3*+ . . . +/?*- 1 ) into (1+0+0+ . . . +(!)), so that this product still remains equal to zero. Similarly we put in place of r 2 of the roots y, y 2 , . . . , y r ~ l , and 1 for the remaining root, and so on. Proceeding thus, we shall ultimately change (4) into an equation of the form where A" contains roots of the form =*=c p only. ADVANCED THEORY OF LINEAR GROUPS 87 Finally, we put 1 in the place of every root a, a 2 , . . . , aP" 1 , as well as every root e p . The left-hand member may then no longer vanish, but will in any event become a multiple of p. The final value of the expression (5) would be (w + l)(l-|-l) = 2 or 0, according as w is replaced by or 1. Notation 1. Any expression N which is a sum of roots of unity changed in the manner described above, shall be denoted by N p . 3. We shall now study the effect of these changes upon the left-hand member of (3). Each of the char acteristics [VS], . . . , [F$ M ], being the sum of three (unknown) roots of unity, will finally become one of the seven numbers 0, 1, ==2, 3, whereas [V], being the sum of three roots of index 1 or p (cf. 1), will become 3. The left-hand member of (3) will thus take the form (6) [V&\i+3Kl+L&VSll+MaV&U, and this number is a multiple of p, by 2. The values K p , L p , M p may be obtained by treating them as indeterminates 0/0. Thus, K = 02) (02 i We find (cf. 132) 0-2 a 3 0* and if we substitute in (6) and multiply by p I we obtain the congruence (7) [VS"]l=8p*+tfJi+v (modp), s, t, v being certain integers, the same for all values of /x. We finally substitute in succession ^ = 0, 1, 2, . . , p 1 in the right-hand member of (7). The remainders 88 FINITE COLLINEATION GROUPS (mod p; cf. 129) should all lie between 3 and +3 inclusive, the interval of the values of [VS^p. Now, each of these seven remainders can correspond to only two different values of ^ less than p, if s and t are not both = (mod p) ( 130). Hence, there will correspond to each of the seven remainders at most 14 different values of /*, so that p is not greater than 14 unless s==0. Trying p= 13 and p = 11, choosing for s, t, v the different possible sets of numbers <p (the problem can be simplified by special devices),* we find that in no case can the re mainders all be contained in the set 0, 1, 2, 3, unless s=2=0. Choosing therefore this alternative we get, if p>7, [V&]i=v (mod p). In particular, v=[V\i = 3 (modp), from which it follows that [VS] P = 3. Again, from this equation we deduce that the roots of [VS] are of index 1 or p. For if the index of one of these roots were divisible by the square of a prime, or by a prime different from p, then the changes indicated in 2 could be made at the outset in such a way that or 1 would take the place of this root. But then [VS] P would be one of the numbers 0, 1, 2, -3. 4. Accordingly, the product VS of any two trans formations both of order p is a transformation of order p or 1. The totality of such transformations in G, together with E, will therefore form a group P ( 27). The order of this group must be a power of p since it * Since an +b runs through the p values 0, 1, 2 ..... p 1 (mod p) when /* does, we may substitute this expression for M in the right-hand member of (7) and select the constants a, b so that this member takes a simpler form. For instance, if p =11, the right-hand member of (7) may bo reduced by this substitution to one of the forms M 2 +c; M; ore; accord ing as 8*0; s=0, <^0; s=t=0 (mod p). When p =13 we get the forms ; c. ADVANCED THEORY OF LINEAR GROUPS 89 contains no transformation whose order differs from p or 1. Moreover, P is invariant under G, since an operator of order p is transformed into one of order p (Exercise 2, 30). Hence, G has an invariant subgroup P of order p k . But this subgroup is abelian (61, Corollary), and therefore G is intransitive or imprimitive (Lemma, 61). 64. Case of n variables. Applying the process indi cated in the previous paragraph to a group Ginn variables, we select any two transformations S, V, both of order p, and write S in canonical form S=(ai, 2, . . . , ). Assuming that the multipliers a 1} . . . , a n are all dis tinct, we find the following congruence corresponding to (7) (8) [VS]^sn n - l +tn n - 2 + . . . (modp). There are at most 2n-fl different values that [VS*] P can take, namely 0, =*=!, = fc 2, . . . , =*=n; and each cor responds to at most ra 1 of the p values of /A: 0, 1, 2, . . . , p 1, if the right-hand member is not merely a constant. It follows that at most (2w+l)(n 1) of these p values of /* can satisfy the congruence (8). Ac cordingly, if p>(2n-fl)(n 1), the right-hand member of (8) must be merely a constant; and then [V&]l=[VS\l=[V]l = n (modp), from which we find [VS]p = n. Now we deduce, as in 4, 63, that the transformation VS is of order p or 1, and from this that G has an invariant abelian subgroup of order p k and is therefore not primitive. Hence the THEOREM 5. The order of a primitive linear group in n variables is not divisible by a prime number which is greater than (2n-\-l)(n 1). 90 FINITE COLLINEATION GROUPS In the case n = 4 this limit is 27. But we may try out the congruence (8) in detail for the primes p = 23, 19, . . , with the result that the right-hand member must reduce to a constant when p>!3. For p=l3 or 11 the process fails, as congruences of the form (8) exist for these primes (for instance, when p=l3, (8) is satis fied if the right-hand member is the function 2/x 3 ). EXERCISES 1. No transformation of variety m (cf. 65) and of order p, a prime greater than (2n-f-l)(ra 1), can belong to a primitive group. 2. No primitive group G can contain a transformation of order p 2 , if the prime p is greater than n ( 67, Corollary). Hence prove that G cannot contain a subgroup of order p 2 , if p=(2n + l)(n 2) and >2. 65. The variety of a linear transformation and of an abelian group. If the number of distinct multipliers of a transformation written in canonical form is k, we say that the transformation is of variety k. Thus, the trans formation S=(l, 1, 1) is of variety 2. By an obvious extension to abelian groups written in canonical form we say that such a group is of variety k if its variables may be separated into k sets in the manner explained in 61. The group H of the illustration in that paragraph is accordingly of variety 3. Notation 2. An expression N which is the sum of a certain number of roots of unity, in which every root c p is replaced by 1, but in which none of the other changes indicated in 2, 63, are carried out, will be denoted by N p . IfN = Q, then N p =0 (mod p). 66. Theorem 6. Let a group G contain a transforma tion S of variety k and order p a <f>, where a>l and p a ^kp; p being a prime number. Then there is an invariant sub group H p in G (not excluding the possibility G = H P ) which ADVANCED THEORY OF LINEAR GROUPS 91 contains S pa ~ l . Any transformation in H p , say T, has the property expressed by the following congruence: (9) [V] P ~[VT] P (modp), where V is any transformation of G. The proof follows the plan of the proof of Theorem 4, 63. We write S in canonical form, and construct the products VS, VS 2 , . . . , VS k ~ l , VR; R denoting S pa ~ l . As in 1, we obtain an equation corresponding to (3): [VR]+K[V]+L[VS]+ . . . However, the changes indicated in 2 are not carried out except that 1 is put for every root e p whose index is a power of p (cf. Notation 2). All the coefficients L, . . . , X become multiples of p by this change (132), and we find K p = 1 (mod p). Hence finally, (10) [VR] P -[V] P =Q (modp). Now consider all the conjugates Ri, . . , Rh to R within G. They generate an invariant subgroup H p (Exercise 3, 31), and they all have the property of R as expressed by (10), since they fulfil the conditions of the theorem (cf. 86, 3). Moreover, any transformation T in H p satisfies the congruence (9), since such a trans formation can be written as the product of powers of #1, . . . , R h . For instance, let T = R 1 R 2 . By (10), we have [F/y p =[F] p , and [(VRi)Ri] p =[VRi] p (mod p). Hence, [F/y p =[F] p (mod p). 67. On the form of the group H p . Let T be any transformation of H p . Then by (9), [] = (modp), 92 FINITE COLLINEATION GROUPS and therefore, since T m belongs to H p also, [T m ] p =n (mod p) . Now let the order (say t) of T be prime to p, and let us assume that a of the multipliers of T are equal to each other, then b of the remaining multipliers equal, and so on. That is, m = a"i+&2+ . . = [T] P , a+b+ . . =n, and [T m ] = 0a m : + ba r ? + . . =[T m ] p . Then we have ( 133, 2, 5) t t m = 1 m = 1 = or = nt, according as a^ 1 or 04. = 1. Hence, we must have correspondingly a=0 or =n (mod p). Like results hold for the other numbers 6, ... If p>n, the congruence a=0 (mod p) is impossible unless a = 0. It follows that every root a s is unity. In other words, H p can in this case contain no transformation whose order is prime to p. The order of H p is therefore a power of p, and G is not primitive (cf. 61). Hence the COROLLARY. No primitive group in n variables can contain a transformation of order p 2 , if p is a prime number greater than n. Remark. The group H p as defined above may very well coin cide with G. However, it may be shown that if a primitive group G contains a group H P , then the latter (or a modified form of it) does not make up the whole of G, and n must be divisible by p*. EXERCISE No primitive group in n variables can contain a transformation of order p 3 , if p is a prime greater than n/2. * Transactions of the American Mathematical Society, XII (1911), 39-41. ADVANCED THEORY OF LINEAR GROUPS 93 68. Theorem 7. Let G contain two commutative transformations S=(ai, . . , a n ) and T=(fti, . . , /?), satisfying the following conditions: (A) Their orders are respectively m$ and p<f>, p being a prime number which is not a factor of m; (B) the variety k of S is equal to or less than m; (C) whenever a s = a. t , then shall (3 s = Pt (the converse is not implied). Under these conditions G has an invariant subgroup H p which contains the transformation T. Proof. Let F = [a s< ] be any transformation of G, and let the products VS, VS 2 , . . , VS k ~ l , VT be con structed. Eliminating an, 022, . . , a nn from the equa tions F = an+022+ . . (cf. (1), 63), we obtain an equation corresponding to (2), with the elements [F], [VS], . . , [F/S*- 1 ], [VT] in the first column. We now write 1 for every root c p whose index is a power of p, before expanding the determinant. The following changes will take place: the elements in the first column [F] . . become [V] p . . (cf. Notation 2, 65), and the elements of the last row* excepting [FT] P , all become unity. Sub tracting finally the last row from the first and expanding, we get ((V] P -[VT] P )D=0 (modp), D being the product of the differences of the k distinct roots among a lt . . , a n . Now, since S m is the identity or a similarity- transformation, a^= . . =a. Hence, there is a rationalizing factor of a s a t such that the product is m or a factor of m. (For, put a t /a s = ) say, a root of index mi (a factor of m), and we have ( 133, 5) (1-0) . .(l-0 m - 1 )=mi*). The product in question *Thc value of lira - ^- . 94 FINITE COLLINEATION GROUPS is therefore a number prime to p. It follows that the product of D and a certain rationalizing factor is an integer N, prime to p. The resulting congruence may now be multiplied by an integer N such that NN =1 (mod p, 131), and we obtain [V] P -[VT] P =0 (modp). Finally, the existence of an invariant subgroup H P} containing T, is proved as in 66. Another useful theorem based on the same principle as the last two is the following : // a primitive group G in n variables contains an abelian subgroup of order p a <J> and variety k, where p is a prime, and a=/c, then G contains an invariant subgroup Hp which does not make up the whole of G, and p must be a factor of n. Cf . reference given in Remark, 67. EXERCISES 1. Prove that no primitive group in four variables can contain two commutative transformations as follows: one of order p = 5, 7, 11, or 13; the other of order q, prime to p, and of variety 4. 2. Under what conditions may two commutative transforma tions of different prime orders p and q belong to a group which con tains no invariant subgroups H p or H q ? 3. Let a group G in four variables contain an abelian subgroup generated by the group E, S- 2 = (l, 1, -1, -1), & = (!, -1, -1, 1), <S 4 =(1, 1, 1, 1), and the transformation T = (l, 1, w, w 2 ), where w 3 = l. Prove that G contains an invariant subgroup of order 3 , involving T. 69. Unit-circle. In the complex plane, a root of unity a. = x+iy = cos 0+i sin represents a point on the unit- circle x 2 +y 2 =l. Accordingly, the multipliers of a trans formation $=(ai, 02, . . , a n ) will represent, m points on the unit-circle if S is of variety m. Thus, the multi pliers of S = (i, i) represent two such points, separated by an arc of 180. ADVANCED THEORY OF LINEAR GROUPS 95 We find it convenient in this and the next few para graphs to indicate the matrix of a transformation S = [a st ] by its first row : S = [an a^ . . . . ai n ] . LEMMA 1. Let S=(a 1 , . . , a n ) be a transformation in canonical form, whose multipliers are distributed over an arc A of the unit-circle of less than 18GP, and let T be a unitary transformation ( 19). Then no element in the principal diagonal of the matrix of the transformation V= TST~ l can vanish. Proof. There being no loss of generality by limiting ourselves to the case of three variables, we put T = [an #12 #13]. Then ( 19) T~ 1 = [du 021 031], and we find TST- 1 = [6n 612 6i 3 ], where ^ S = a,ia sl a 1 +a, 2 a 6 . 2 a 2 +a,3a s3 a3 (=1, 2, 3). Denote the number a st a st (which is real and positive unless a s , = 0) by p t . Then Now, if we regard a. 1} a^, a 3 as unit vectors radiating from the center of the unit-circle to points of the arc A, the sum b ss will evidently be a non-vanishing vector lying in the angle subtended by the arc A. Hence the lemma. It is furthermore evident that the length of the vector b ss is less than the sum of the lengths of the vectors pi^, ^2^2, ??3 a 3 (namely ^1+^2+^3 = 1), unless those of the vectors just mentioned which do not vanish have the same direction. Hence the LEMMA 2. No element b ss of the matrix of V of Lemma 1 can be a root of unity (or even be of unit length} unless those of the vectors p\&\, . . . , Pn a n which do not vanish, have the same direction. Hence, if b ss is a root of unity, and if neither a st nor a sv vanish, then a t = a,. 96 FINITE COLLINEATION GROUPS 70. Theorem 8. // a group G contains a trans formation S=(a 1} . . . , a n ) whose multipliers, when located on the unit-circle, occupy an arc ^=120 and extend ing not more than 60 on either side of some one of them, say aij * then G is not primitive. (It is assumed that S is not a similarity-transformation.) Let n = 3 to begin with. We assume that G has the unitary form and that at the same time S has the canonical form (Corollary, 22). Furthermore, we assume for the present that (11) Ctl4=tt2, a l=F a 3, leaving to the end of 71 the treatment of the general case. We shall prove in order (A) The transformation S and all its conjugates in G generate an intransitive group G (71). (B) Due to the invariance of G (Exercise 3, 31), the group G cannot be primitive ( 72). 71. Proof of (A). Since the multipliers a lt a 2 , a 3 all lie on an arc ^120, no element in the principal diagonal of any conjugate of S can vanish (Lemma 1, 69). Hence, if V=[bn 612 613] is such a conjugate, then &n=|=0. Now, if 612 = 613 = for every conjugate, the group G generated by these conjugates is reducible and therefore intransitive ( 20). The proposition (A) is therefore established by proving 6 12 = 6 13 = 0. To this purpose let us denote by M the aggregate of all those conjugates of S, if any, for which at least one of the two elements 612, bu does not vanish. * For an illustration, take the transformation of order 6 in three variables whose multipliers are a l =1, a 2 = w, a 3 = u>, where w=l; or one of order 7 whose multipliers are a l = 1, a a =cos v -f i sin v, a 3 = a 2 i = cos v i sin B, where v =(360/7). ADVANCED THEORY OF LINEAR GROUPS 97 Let T = [a/! a{ 2 a^] be any one of these. We then introduce a f unction f(T ) denned as follows: We have (12) since ai ai +a^ a/ 2 +a^ d^ = 1 (if a^ a^ = 1, it would of necessity follow that a{ 2 = a{a = 0, contrary to the hypoth esis concerning M). We now let T = [a st ] be that transformation in the aggregate M for which f(T) has the greatest value. Con structing Ti = TST~ l we shall then demonstrate the following properties of T\: (1) /(7\) >f(T) , * (2) T, belongs to M , thus violating the assumption as to T. The conclusion is drawn that M does not exist. To prove (1), put 02/04 = cos v z -\-i sin v 2 , a 3 /a 1 = cos v z +i sin v s , and we find where cos t>2+ai3i3 cos Now, by the condition of the theorem, s = 2, 3), so that cos v s * The use of an inequality of this kind is in substance a discovery of Valentiner s (De endelige Transformations-Gruppers Theori, Copenhagen, 1889, p. 115). An equivalent inequality is used in an ingenious manner by Bieberbach to find a limit to the orders of linear groups. (See Sitzungsberichte der Kgl. Preussischen Akademie der Wissenschaften, 1911, pp. 231-40; also two papers by Probenius in the same Proceedings (1911) pp. 241-48 and pp. 373-78. 98 FINITE COLLINEATION GROUPS Hence, we derive \)=A^ aiidn + i (ai2i2 + i3i3) = iian + i ( 1 nan) Accordingly, since 1+X 2 >2A unless A = 1, by (12), it follows that To prove (2), we first notice that T^TST 1 is a conjugate of S. It remains for us to show that, if TI = [6n 612 613], 612 and 613 do not both vanish. Assume the contrary : 612 = 613 = 0. Then bu&u = 1, so that the vector bn is of unit length (in fact, 6n must be a root of unity). But, and therefore, since an=}=0 (Lemma 1, 69), and since by assumption either 0,12 or ctia^O, say a^^O, we have ai=a 2 (Lemma 2, 69), contrary to the hypothesis (11). Hence TI belongs to M if T does. In the general case, the multipliers of S may be repeated. Assume that a h a 2 . . . occur respectively ra, k, . . . , times: S = (a.i, . . . , aij a 2 , . . . , a 2 ; . . . ) . The aggregate M consists here of all those conjugates V = [b s t] of S for which the equations (13) b st = (s = l, 2, . . . , ra; t = m + l, ra+2, . . . , n) are not all satisfied: As above, the proof of (A) depends on show ing that such an aggregate does not exist; and to this end the func tion }(T ) is introduced, which now has the form f(T ) = + (o{i +aij+ . . . +a mm )(ai\ In place of (12) we here have f(T )<m. The inequality (1) *(Ti)>f(T) is proved without difficulty. ADVANCED THEORY OF LINEAR GROUPS 99 To prove (2) we assume first that Ti = TST~ l does not belong to M; that is, if Ti = [b s t], we assume the set (13) true. Now we write TI in canonical form as far as the first m variables are con cerned; S will still have the canonical form, and G can still be written as a unitary group. Moreover, the new aggregate M will coincide with the old. If then TI = (0 lf 2 , . . , Om) as far as the first m variables are concerned, we have m+i-f- . . dm By using the Lemma 2, 69, and the facts that T = [a s t\ belongs to M and that none of the numbers an, a 2 2, . . , a mm vanish, we prove readily that at least one of the system of equations just written is impossible. Hence T\ belongs to M. 72. Proof of (B); general case. By (A), the first m variables x\, . . , x m are transformed into linear func tions of themselves by S and its conjugates, if ai is repeated m times among the multipliers of S. The group G f generated by these conjugates is therefore reducible, and hence also intransitive; say in the two sets X=(xi, . . , x m ), Y=(x m+ i, . . , x n ) to begin with. Now let G f be broken up into its ultimate sets of in- transitivity in any way whatsoever, without reference to the original intransitive set X. Then if y\, . . , yk is a set of intransitivity, and if S is written in canonical form as far as this set is concerned, its multipliers within the set are all =a it or none of them are. (That is, in the first case the variables are linear functions of x\, . . , x m ; in the last case they are linear functions of m +i, . . , x n .) This is easily seen when y\ y . . , yk are expressed as functions of x\, . . , x n , and the condition imposed that S transforms these functions into constant multiples of themselves. Thus, we find that if / multipliers are equal 100 FINITE COLLINEATION GROUPS to ai, and the remaining multipliers are not, then / vari ables, say 2/1, . . , 2//, are functions of x\, . . , x m only, and the remaining variables 2/7+1, , Vk functions of x m+ i, . . , x n only. Since these two sets make up a single intransitive set for G , and the sets X, Y are sepa rately intransitive sets, it follows that (2/1, . . , 2//) and (2//+i, . . , 2//0 must make up two intransitive sets. What has just been said with regard to S will be true of any conjugate of S within G, since such a conjugate might have been chosen for S in the first place (conjugate transformations have the same multipliers; cf. Theorem 11, 23). Hence, as now written, G possesses the prop erty that any one of the conjugates of S will appear partly in the canonical form, namely in regard to the m variables that correspond to its m multipliers ai. Now, a given set of intransitivity may have the prop erty that several conjugates Si, . . , S p have the canoni cal form (ai, . . , ai) for this set. Calling p the " index" of the set in question, we seek the sets of highest index -*. Then we can show that G permutes these sets among them selves, thus proving (B). Let (zi, . . , Zh) be such a set, and let Si, . . , S* be the corresponding conjugates. Selecting any trans formation Fin G, we write V^SiV^ 7\, . . , V~ 1 SnV = T n , and (zi)V = vi, . . , (z h )V = v h . Then we have, (z lt . . , z h )SjV=(zi, . . , z h )VTj (j=l, 2, i.e.. The variables v\, . . , v h are therefore transformed into <*-iVi } . . , aiVh by the maximum number of distinct con jugates, namely T\, . . , T n . It follows that v\, . . , Vh are linear functions of the variables of a single set of index TT. For, it is easily seen that if a certain linear function ADVANCED THEORY OF LINEAR GROUPS 101 v of the variables of G is transformed into < & by tKo maximum number of conjugates, v must, of; nectssity be a linear function of the variables of sorrte* ont v of ttttf sets of index v. 73. Theorem 9. If a linear group G contains an abelian subgroup K of variety m ( 65) and order k<f>, where /^ 6 m-i_( 6 m- 2 _|_ 6 m-3_|_ +6) , then G is intransitive or imprimitive. We can prove that there is in K a transformation all of whose multipliers are not more than 60 removed from some one of them when plotted on the unit-circle. Theorem 8 may then be applied. Let K be written in canonical form, and let the vari ables be arranged in m sets according to the method explained in 61. The case n = ra = 4 will be discussed in detail; it will then be sufficiently obvious how to prove the theorem for the general case. Furthermore, since only the mutual ratios of the multipliers are of importance in Theorem 8, a slight simplification can be effected by adopting the form $ =(1, /?/<*, y/ a , ) instead of S= (a, /?, y, . . ). Thereby we gain an additional advan tage: the order of the group composed of the transfor mations S r , . . is k when the order of the corresponding group S, . . is k* ( 51, 3). Accordingly, we have fc^6 3 (6 2 +6) distinct trans formations (14) S t = (l,a t ,p t , yt ) (f = l,2, . . ,fc). We now plot the points 04, a 2 , . . , a^ on the unit-circle. Starting at one of these points, we divide the circle into 6 equal parts. It is evident that one at least of these parts will contain more than k/Q points, either within or upon its boundary. (In the case under discussion, one part will contain at least 6 2 6 points.) There are 102 FINITE COLLINEATION GROUPS therefore G 2 Dr more points which are not separated one frotn another by more than 60. Let the corresponding iransforniatio ris of the set (14) be denoted by Si, Sz, . . , /Sao. We next plot the points ft, /3 2 , . . , fto on the unit- circle. Dividing this circle as before, we find at least 30/6+1 points lying on one of the arcs of 60, correspond ing to the transformations Si, Sz, . . $ 6 . We finally plot the points 71, . . , ye and find at least 2 points on an arc of 60. If the corresponding transformations are Si, Sz, it is plain that their corresponding multipliers are never more than 60 apart. It follows that the trans formation $2$T 1= (1, at/a-i, ft/A, 72/71) fulfils the con ditions imposed on S in Theorem 8, the root a x there being unity here. NOTE. By modifying to some extent the principle developed in 69-72, and by using certain processes in the Geometry of Numbers (cf. Transactions of the American Mathematical Society, XV (1914), 227), the author has obtained a very much lower limit than the one given in Theorem 9; namely /c^A^- 1 , where h is a number lying between 3 and 5, depending on the nature of the prime factors involved in the order of the group K. The proof of this result has not yet been published. EXERCISES 1. In a group G, a transformation all of whose multipliers lie on an arc < 60 is commutative with one whose multipliers lie on an arc < 180 (Frobenius). 2. Prove that if a group G contains a transformation T all of whose multipliers lie on an arc < 72, then G is not primitive. (Hint: Either Tor T 5 fulfils the conditions imposed on S of Theorem 8.) 3. By Theorem 9, a primitive group G cannot contain an abelian subgroup of variety 3, whose order k<f> is equal to or greater than 300. Examine in detail the possibilities fc = 26, 27, 28, 29, and prove that 74. Order of a primitive group in n variables. We are now in a position to set a superior limit to the order g<t> ADVANCED THEORY OF LINEAR GROUPS 103 of a primitive group G in n variables. A Sylow subgroup of order p a is monomial and must contain an abelian sub group of order p a ~ b at least, where p b denotes the highest power of p which divides n\ (Exercise, 61). Hence, by 73, Note, and that part of g which is made up of prime factors not greater than n is not greater than w!5 (n ~ 1)fl(n ty, where 0(ri) denotes the number of primes smaller than n+1. Again, if n-f-1 is a prime and is a factor of g, the cor responding Sylow subgroup is abelian (Corollary, 61) and its order is ^5 n-1 . Finally, the remaining factor of g is the order of an abelian group ( 62) and is also ^5 n-1 . It follows that the order of a primitive group in n variables is HISTORICAL NOTE. In 1878 Jordan proved the classi cal theorem that the order of a finite linear group in n variables is of the form A/, where/ is the order of an abelian self-conjugate subgroup, and where A. is inferior to a fixed number which depends only upon n.* Definite limits to A have been given by Schur for the case where the characteristics belong to a given algebraic domain ;f by Bieberbach and Frobeniust and by the author. Concerning the theorems of 60-68, see Transactions of the American Mathematical Society, IV, 387-97; V, 310-20; VI, 230-32; VII, 523-29; XII, 39-42. * Journal fUr die reine und angewandte Mathematik, Bd. 84, p. 91. t Sitzungsberichte der Kdniglich-Preussischen Akadcmie der Wissen- schaften, 1905, pp. 77 fl. t Sitzungsberichte, etc., 1911, pp. 231 ff., 241 flf. Transactions of the American Mathematical Society, V (1904), 320-21 for primitive groups; VI (1905), 232 for imprimitivo groups. CHAPTER V THE LINEAR GROUPS IN THREE VARIABLES 75. Introduction. The determination of the linear groups in three variables is here based on the following classification : 1. Intransitive and imprimitive groups. 2. Primitive groups having invariant intransitive or imprimitive subgroups. 3. Primitive groups whose corresponding collineation groups are simple (" primitive simple groups"). 4. Primitive groups having invariant primitive sub groups. No specific theory is needed for the determination of the groups in the first class, beyond that given in 60. For the determination of the groups in class 3 the theorems of 61-73 are very useful. The notation and conventions laid down in 51 are observed throughout the chapter. 76. Intransitive and imprimitive groups. There are two types of intransitive groups: (A) Xi=ax{, xz = fix!>, x 3 = yx^ (abelian type). (B) xi = i In (B) the variables z 2 , x 3 are transformed by a linear group in two variables (cf. chap. iii). The imprimitive groups are all monomial. There are two types: 104 LINEAR GROUPS IN THREE VARIABLES 105 (C) A group generated by an abelian group H: Xi = ax[, X2 = Px?>, Z 3 = y4 and a transformation which permutes the vari ables in the order (xiX 2 x 3 ). This transforma tion may be written in the form T: xi = x?,, X2 = x t, X3 = x[, by a suitable choice of variables. (D) A group generated by H, T of (C) and the trans formation R: x\ = ax[, #2 = 6x3, x s = cxi. 77. Remarks on the invariants of the groups (C) and (D). Interpreting xi, x z , x s as homogeneous co-ordinates of the plane, the triangle whose sides are Xi = Q, #2 = 0, z 3 = is transformed into itself by the operators of (C) and (D) ; in other words, XiX^x^ is an invariant of these groups. Later on it becomes necessary for us to know under what conditions there are other invariant triangles. Assuming the existence of one such, say ( 1 ) we operate successively by the transformations of H and by T. Observing that a, /?, y cannot all be equal for every transformation of H , as otherwise (C) and (D) would be intransitive, we find by examining the various possi bilities that (1) could not be distinct from XiX 2 ^3 = unless H is the particular group generated by the transformations & = (!, w, <o 2 ), 82 = (w, o>, w) ; u> 3 =l. 106 FINITE COLLINEAT1ON GROUPS There are then four invariant triangles for (C), namely : = 0; (2) (0=1, CO, Or 0)2). In the case of (D), these four triangles will be invariant if the group is generated by Si, S 2 , T and R, the latter now having the form R: x 1 =-x[ , either directly or after multiplication by suitable powers of Si and S 2 . 78. Groups having invariant intransitive subgroups. A group having an invariant subgroup of type (A) is intransitive or imprimitive (Lemma, 61), and a group having an invariant subgroup of type (B) is intransitive. For, let V be any transformation of such a group, and T any transformation of (B). Then VTV~ 1 =Ti belongs to (B), and if we put (xi)Ti = ax 1} (xi)V = y, we have This shows that y = Q is an invariant straight line for (B). But xi = is the only such line, and therefore y = (xi) V = kx\, k = constant. The group in question is therefore reduc ible and hence intransitive. 79. Primitive groups having invariant imprimitive subgroups. We now consider a group G containing an invariant subgroup of type (C) or (D), 76. These types leave invariant the triangle Xix 2 x 3 = (77), and if this is the only one, we could prove by the method of 78 that G would also transform this triangle into itself. But then G would not be primitive. We therefore assume LINEAR GROUPS IN THREE VARIABLES 107 that there are four invariant triangles for (C) and (D), permuted among themselves by the transformations of G. Let us denote the triangles by h, k, t 3 , U, in order as they are listed in (2). We now associate with each transformation in G a substitution on the letters ti, k, t 3 , h, indicating the manner in which the transformation permutes the corresponding triangles. We thus obtain a substitution group K on four letters to which G is multiply isomorphic (32), and the invariant subgroup (C) or (D) corresponds to the identity of K. No one of the four letters could be left unchanged by every substitution of K. For the corresponding triangle would be invariant under G; and bringing this triangle into the form XiXzX 3 = by a suitable choice of new variables, G would appear in intransitive or imprimitive form. Moreover, no transformation can interchange two of the triangles and leave the other two fixed, as may be verified directly. Under these conditions we find the following possible forms for K: (E ) 1, (MO (W;* (F ) 1, (to) (WO, (MO (MO, (MO (WO; (G ) the alternating group on four letters, generated by (JifeXWO and (Ws0. Now, to construct the corresponding linear transforma tions we observe first that the group (D) as given in 77 contains all the transformations which leave invariant each of the four triangles. Next we note that if a given transformation V permutes the triangles in a certain manner, then any transformation V which permutes them in the same manner can be written in the form V = XV, X being a transformation of (D). For, V V~ l must leave * The throo types of G corresponding to the three substitution groups : E, (/!*,) (* 3 f 4 ) ; K, (^j) (< 2 I 4 ) ; E, (V 4 )(V3> aro equivalent (51). 108 FINITE COLLINEATION GROUPS fixed each triangle, and is therefore a transformation X as defined. We are now in a position to construct the required groups. By direct application we verify that the trans formations U, V, UVU~ l : U: Xi = V: xi = UVU- 1 : permute the triangles in the following manner: Accordingly, since all the groups required contain a transformation corresponding to (fib)(J&)) every such group must contain a transformation XV, X belonging to (D). Hence, if G contains (D) as a subgroup, it also contains V. If, however, (C) were a subgroup of G, but not (D), then either V is contained in G, or else XV, where X is a transformation contained in (D) but not in (C). In this event X may be written X\R, where X\ belongs to (C). Hence, finally, either V or RV belongs to G. Howpver, V 2 =(RV) 2 = R. Thus R, and therefore also V, are contained in G in any case. Again, if G contains a transformation corresponding to (2^4) or (t\t^(tzU), such a transformation can be written XU or XUVU- 1 , X belonging to (D). Hence, since G contains (D) as we have just seen, it will contain either U or UVU 1 in the cases considered. We therefore have the following types: LINEAR GROUPS IN THREE VARIABLES 109 (E) Group of order 36< generated by (C) as given in 77: and the transformation V of (3). (F) Group of order 72< generated by Si, T, V and UVU~ l . (G) Group of order 216< generated by &, T, V and [7. These groups are all primitive, and they all contain (D) as an invariant subgroup. The group (G) is called the Hessian group.* 80. Primitive simple groups :f the Sylow subgroups. In order to utilize Theorems 4-7, chap. IV, it becomes necessary for us to study the effect of the presence in a group G of an invariant subgroup H p ; or, since G is here to be a simple group, \ to determine the possibility G = H P . Now, it is at once seen that no transformation T= (ai, a 2 , a 3 ) whose order is prime to p, can belong to H p , by the results of 67. Accordingly, the order of H p is a power of p. However, a group of this order is not primi tive (61), and therefore we shall dismiss from considera tion in 80-81 all groups which may be shown to contain H p , as for instance a group containing the transformation T=(lji t i), where i = V 1 (cf. 66), or the transforma tion T r =(o>i, cw 2 , 2 w 3 ), where a>5 = co:j = G>jj=l, and where e is a primitive 9th root of unity. On the other hand, the presence of the transformation T=(e<i, eo> 2 , eo> 3 ) does not imply the presence of a subgroup H p . * Cf. Jordan, Journal fur die reine und angewandte Mathematik, Bd. 84 (1878), p. 209. t Such a group may contain the group of similarity-transformations, which is an invariant subgroup (cf. 51, 1). t If G contains the group of similarity-transformations, we can assume that //p does so too. 110 FINITE COLLINEATION GROUPS We now proceed to enumerate the different possible types of Sylow subgroups. In this list, the symbol P& means " Sylow subgroup of order ft." p = 2: (a) P 2 , generated by (1, -1, -1). (b) P 4 , generated by (1, -1, -1) and (-1, 1, -1). (c) P 4 , generated by (1, i, i). (d) P 8 , generated by (6) and T f : (e) PS, generated by (c) and T . (a) P 3 , generated by (1, w, o> 2 ) . (b) P 3 ,j>, generated by (e, c, ew 2 ), where c 3 = o>. (c) Pg,*,, generated by (a) and (6). (d) PW, generated by (a) and T": x\ = xj, x 2 = zj, x 3 = x[. (e) Pzit, generated by (6) and T". A group of order 5 a is abelian and contains no trans formation of order 5 2 ( 67, Corollary). If therefore a =2, we necessarily have two or more generating trans formations, say $i = (ai, a 2 , a 3 ) and >S2=(a 4 , a 5 , a 6 ), both of order 5. If now a represents a primitive 5th root of unity, a transformation of order 5 and variety 2 can be found in G, namely T = SfS* = (a, a, a~ 2 ). (We merely determine a and b such that aa^ = a^a| .) The transformation T leaves invariant every straight line through the point X: Xi = x 2 = 0. If R be any con jugate to T, then R would likewise leave invariant every straight line through a certain point Y. Accordingly, both T and R leave invariant the straight line which joins LINEAR GROUPS IN THREE VARIABLES 111 X and Y. Let the variables be so changed that this line is i = 0; the group generated by T and R will then be intransitive ( 20). If the component H of this group in the variables x%, x 3 is not primitive, T and R are com mutative, because they will appear in the canonical form. Therefore, either, 1 all the transformations conjugate with T are mutually commutative, or, 2 for at least one pair of such transformations (say T and R) there is a primitive group H in two variables (x 2 , x 3 ). This group must be of type (E), 58, and contains the follow ing transformation ( w, w 2 ). Therefore G contains the transformation (1, o>, w 2 ), and is not primitive ( 70). In the alternate case 1, G contains invariantly the abelian group generated by T and its conjugates and is not primitive (61). Accordingly, a=l, and a transformation of order 5 must be of variety 3. By trial, we now find the following type: P 5 , generated by (1, a, a : ), ;- 5 1 . A Sylow subgroup of order 7 a is abelian. By 73, a<2, and by 70, we can have no transformation of type (a 2 , a 2 , a 3 ) or (1, a, a~ ! ), where a 7 =l. Accordingly, we are limited to the following type: P 7 , generated by (ft /8 2 , /?<); (?= 1 . The types P 5 , PI are mutually exclusive. For, if both were present in a group, we should have a transforma tion of order 35 ( 90, Exercise 2), and therefore two commutative transformations of orders 5 and 7, namely those listed above under P 6 and P 7 . But this would imply a subgroup H p (68). 112 FINITE COLLINEATION GROUPS For the same reasons, the types (6), (c), and (e) for p = 3 cannot exist simultaneously with a subgroup of order 5 or one of order 7 in a primitive group. 81. The orders of the primitive simple groups. Con sider first the case of a group G of order g<t> = lg\<t>. By 80, gi is a factor of 2 3 . 3 2 </>, and we have 8 or 36 subgroups of order 7 ( 36) in the corresponding collineation group. In the first case, P 7 is invariant in a group H of order h = g<f>/8 ( 30), and in the second case it is invariant in a group of order 0</36. Now, PI is generated by a trans formation (ft, /3 2 , /3 4 ) as we have seen; it is therefore a simple matter to show that the groups of orders 63< or 14< would be impossible or would imply a subgroup H p . It follows that = 252 or 168. Consider next a group G of order g< = 5</i</>, g\ being as above a factor of 2 3 .3 2 . Here we have 6 or 36 Sylow subgroups of order 5. Now, it is easily verified that such a group, generated by (1, a. -~-), cannot be invariant in a subgroup H of ovlor h = 2Qk or 15fc; it follows that the ni^bcrs gr/6 and gr/36 are either 5 or 10; i.e., g is one or the numbers 30, 60, 180, 360. The orders to be considered are therefore 60<, 360< and 168< (cf. 48). There are no simple groups of order 2 a .3 6 (100). 82. The three types of primitive simple groups. The simple groups of orders 60 and 360 are simply isomorphic with the alternating groups on 5 and 6 letters respectively. To determine the corresponding linear groups, we con struct the transformations F\, F 2 , . . , satisfying the relations specified in 50, as was done in chap. Ill in the case of the group (E). We find the types:* * Of. Maschke. Mathcmatische Annalen, Bd. 51 (1899), pp. 264-67. LINEAR GROUPS IN THREE VARIABLES 113 (H) Group of order 60, generated by F\, F 2 , F 3 , satis fying the relations: p = FI=FI = E, (F 1 F 2 Y namely : Fi: X! = x!>, x 2 = x^, x 3 = F 2 = (l, -1, -1); where ni = %(-l + l/S), /*2 = J(-l-l/5) . (I) Group of order 360<, generated by Fi, F 2 , F 3 of (H) and F, where F 4 : xi=-x, x 2 =-o>X3, x 3 = Concerning the simple group of order 168, the generat ing relations can best be obtained by examining the sub stitution group on 7 letters simply isomorphic with the group in question. * We shall here select the substitutions S=(abcdefg), T=(adb)(cef), R=(ab)(ce)\ satisfying the relations : (J) Group of order 168 generated by S, T, R: T: Xi = x 2 , x 2 = X3, x 3 = x[ R: Xi = where * = !, a = ^3 4 -/3 3 , b = P 2 -p, c = p-p G ; * Cf. Burnsido, Theory of Groups of Finite Order, 2d ed., Cambridge, 1911, pp. 218, 310; Miller, Blichfeldt, and Dickson, Theory and Applica tions of Finite Groups, New York, 1916, p. 50. 114 FINITE COLLINEATION GROUPS 83. Primitive groups having invariant primitive sub groups. A possible subgroup H p is monomial (80). We have already determined the primitive groups con taining a group of this type invariantly; it is therefore unnecessary to consider the groups containing the sub groups H p . Hence, the orders of the groups still to be examined must be factors of the numbers 2 3 . 3 3 <, 2 3 . 3 2 . 5<, 2 3 .3 2 .7<, as we have seen. The group (I) has already a maximum order. Again, the group (J) cannot be invariant in a group of order 2 3 .3 2 .7<, since a possible group of this order must contain a subgroup H p ( 81). As to the group (H), we make use of the fact that its 6 subgroups of order 5 must be per muted among themselves by an assumed larger group (K), in which it is self-conjugate. In any event, we would have a subgroup of order 20k or 30k, containing a given P 5 self-conjugately. But this would imply a group H p (cf. 81). There remain the groups (E), (F), (G). Now, any larger group would permute among themselves the four triangles which form an invariant system for these groups. But this was just the condition under which the three given groups were determined. Hence no new types result. EXERCISES 1. Determine the linear groups in two variables by the method of this chapter. (To determine the primitive simple groups, show first that the order of such a group is a factor of 600.) 2. Obtain the group (H) by the second method given in 58, in the following form : S = (l, e 4 , e); [/ ; Xi= x it x*= -X 3 , X 3 = -X 2 ; T : 3i = where e B = l, s = e 2 - r -e 3 , LINEAR GROUPS IN THREE VARIABLES 115 Show also that the second form of (E), 58, transforms the variables yo= XiX 2 , yi=xl, yi=x\ into linear functions of them selves, and hence this group appears as a linear group in three variables, which is precisely the group (H) as given in this exercise if we write yo, y\, y 3 for x\ t x 2 , x 3 respectively in (H). 3. Obtain the group (I) by adding a transformation W = (ad)(ef) to the list S } U f , T of Exercise 2, and show that W: a;i = where X 1 = i(- 1=^^15), * 2 = l(-l=r\/^l5) . 4. Show that the function xlx 3 +x 2 Xi+x s 3 x 2 is invariant under (J). 5. In 80 it was shown that two transformations, both of variety 2, would generate an intransitive group. By the same process (geometrical), prove that two transformations in four vari ables, both of variety 2, would also generate an intransitive group. Bibliography. Complete discussions of the linear groups in three variables are to be found in the following articles: Jordan, Journal fur die reine und angewandte Mathematik, Bd. 84 (1878), pp. 125 ff.; Valentiner, De endelige Transformations-Gruppers Theori, Copenhagen (Videnskabernes Selskabs Afhandlinger), 1889; Mitchell, Transactions of the American Mathematical Society, XII (1911), 207-42; and the author, Transactions of the Ameri can Mathematical Society, V (1904), 321-25, and Mathe- matische Annalen, Bd. 63 (1907), pp. 552-72. See also Miller, Blichfeldt, and Dickson, Theory and Applications of Finite Groups, 125, where further references will be found. CHAPTER VI THE THEORY OF GROUP CHARACTERISTICS 84. Introduction. This theory, of importance not only for linear groups, but for abstract and substitution groups as well, was initiated by Frobenius and is largely due to him,* though Schurj and BurnsideJ simplified the theory and added extensive applications. We shall devote this chapter to an exposition of the main points of the theory by a process differing in many respects from previous expositions. It is not assumed in this chapter that the transforma tions of a linear group under discussion are necessarily of determinant unity. For the sake of uniformity of notation, we shall throughout this chapter denote the order of a group under discussion by g (unless otherwise specified), however the group may be designated (6r, G , H, etc.) Furthermore, the number of its sets of conjugate operators ( 29) shall be denoted by h, these sets to contain respectively 0i, 02, . . , Qh operators, so that = 0i+02+ . . +gh- * Sitzungsberichte der Kgl.-Preussischen Akademie der Wissenschaften, 1896, pp. 985, 1343; 1897, p. 994; 1899, p. 482. t Sitzungsberichte, etc., 1905, p. 406; Journal fur die reine und ange- wandte Mathematik, Bd. 127 (1904), pp. 20-50; Bd. 132 (1907), pp. 85-137; Bd. 139 (1911), pp. 155-250. t Ada Mathematica, XXVIII (1904), 369-87; Proceedings of the London Mathematical Society, 1904, pp. 117-23; Theory of Groups of Finite Order, 2d ed., Cambridge, 1911, pp. 243 ff. See also the following accounts: Molien, Sitzungsberichte, etc., 1897, pp. 1152-56; Dlckson, Annals of Mathematics, 1902, pp. 25-49; Miller, Blichfeldt, and Dickson, Theory and Applications of Finite Groups, New York, 1916, pp. 257-78; and the author, Transactions of the American Mathematical Society, V (1904), 461-66. 116 THE THEORY OF GROUP CHARACTERISTICS 117 85. Remarks on intransitive groups. Extension of group-concept. 1. Let G=(Si, S 2 , . . , S g ) be an intransitive group, which upon a suitable change of vari ables breaks up into two or more groups, G , G", . . , corresponding to the various sets of intransitivity. We shall have occasion to employ a notation analogous to that used for linear transformations, namely, (POO G" G " and we shall say that G , G", . . are " component" groups of G. 2. Under certain conditions it is convenient to extend the name " group" to a set of operators which are not all distinct, but which can be put into a (1, 1) correspondence with a group G. A group G of order g which is (1, h) isomorphic with G can be exhibited as if it were a group simply isomorphic with G, namely by repeating each of its operators h times. For instance, the substitution group of order 6: E, (ab), (ac), (be), (abc), (acb) is multiply isomorphic with two of its subgroups: E; and E, (ab). With the concept of "group" extended as indicated above, we may exhibit the three groups as simply isomorphic in the following manner: E, (ab), (ae), (be), (abc), (acb); E, E, E, E, E, E; E, (ab), (ab), (ab), E, E. We shall accordingly agree to look upon the groups G, G , G", . . in 1 as simply isomorphic. 86. Characteristics. 1. Definition. Let the vari ables of the group G = (Si, 82, . . , S g ) be x\ t x z , . . , x n . 118 FINITE COLLINEATION GROUPS As noted in 23, the sum of the multipliers of a given transformation St is called the characteristic of S t , and we shall here denote it by xt or x(^)- It is equal to the sum of the elements in the principal diagonal of S t ; if S t is written in canonical form: (a, /3, . . , A.), 2. Inverse and conjugate-imaginary transformations. The multipliers are always roots of unity, and if a, /?, . . are the multipliers of S t , those of S~ l are the reciprocals a" 1 , /J- 1 , . . . Since the reciprocal of a root of unity a is its conjugate-imaginary (a~ 1 = a), we have 3. Conjugate transformations. The characteristics of conjugate transformations are equal ( 23). Hence, if there are h complete conjugate sets of transformations in G ( 29), there cannot be more than h different charac teristics of G. In conformity with the notation adopted in 84, we shall indicate these by the symbols xi, X2> > Xh> 4. Intransitive groups. The characteristic of a trans formation S of the intransitive group G, 85, 1, is evi dently the sum of the characteristics of the component" transformations S , >S", . . : x(S)=x(S )+x(S")+ . . . 5. Substitution groups. When a substitution con sisting of a single cycle on more than one letter, S=(aiOz . . . a m ), is written in matrix form as a linear transformation (1), the elements in its principal diagonal are all zero. Hence its characteristic is zero. If w=l, the transformation has the form S=(l); in this case \(S) = l. The multipliers of *S=(aia 2 . . . a m ) are the m different rath roots of unity; thus, the substitution $ = (0110203), when written as a linear transformation in canonical form, becomes S=(l, <*>, <*) 2 ), where o> 3 = l. THE THEORY OF GROUP CHARACTERISTICS 119 It follows that the characteristic of the most general substitution S on n letters is the integer which equals the number of cycles of one letter; i.e., \(S)=the number of letters that the substitution leaves unchanged. In particular, the characteristics of a regular group ( 47) are all zero, except that \(E) equals the order of the group. Frobenius and Schur call the set of quantities Xi, , Xg a character of G. In their terminology, an abstract group G would possess as many simple characters as there are non-equivalent linear groups to which G is simply or multiply isornorphic (cf. 99). 87. The sum and product of matrices. The sum of a series of square matrices of the same order is the matrix whose elements are the algebraic sums of the correspond ing elements of the given matrices. Thus, 62 a\ 61] [02 ci di.rLca If Si, Sz, . . are linear transformations in the same variables, we shall write &+&+ . . to denote the matrix which is the sum of the matrices of Si, $2, . The product of two matrices is obtained by the rule given in 3, irrespective of whether the matrices represent linear transformations or not. If M represents a matrix, and c a constant or a variable, the symbol cM shall repre sent the matrix obtained by multiplying every element of M by c. In accordance with these definitions we find Si + $2 = . . ) = . )T=T- 1 SiT+T- 1 S*T+ . . , c(Si+S 2 + . . )= 120 FINITE COLLINEATION GROUPS Again, if X represents a linear function of the variables xi, xz t . . , x n , and if the matrix M be regarded as the matrix of a linear transformation (though the determinant of M may vanish), we shall by (X)M represent the result of operating upon X by M (cf. 2). Now if where Si, . . , S m are linear transformations of a group in the variables x\, . . , x nj it is then readily seen that (X)M=(X)S l +(X)&+ . . +(X)S m . 88. Invariants. A function /(xi, . . , x n ) which is transformed into a constant multiple of itself by every transformation of a group G is called an invariant of G. It is an absolute invariant if the constant multiplier is unity for every transformation of G; otherwise it is a relative invariant. A series of invariants /i, . . , f m are said to be inde pendent of each other if the variables cannot be eliminated from the equations where a\, . . , a m are arbitrary constants. They are said to be linearly independent if no identity exists of the form 61/1+ - where 61, . . , b m are constants, not all zero. EXERCISES 1. The substitution group E, (xixd(x& t ), (x#$(x&d,(x is abelian and consequently intransitive when written as a linear group ( 22). Write it in canonical form, and show that it possesses four invariants of the first degree, one of which is absolute. Show also that the characteristics are 4, 0, 0, 0. 2. Write the symmetric group on three letters as a linear group, and construct the matrices which are each the sum of the THE THEORY OF GROUP CHARACTERISTICS 121 transformations of a conjugate set. Then show that any one of these matrices (M) and any transformation (S) of the group are commutative: MS =SM. Show also that the matrices are mutually commutative. 3. Write down the characteristics of the group in the previous exercise, and prove that their sum, the sum of their squares, the sum of their cubes, etc., is always a multiple of 6, the order of the group. 4. Construct the two independent invariants of the first degree of the group of order 6 in 95. Prove also that the alternating group on 5 letters possesses only one invariant of the first degree. 5. Prove that the characteristics of the two transformations AB and BA are equal. ON THE CHARACTERISTICS OF TRANSITIVE GROUPS, 89-91 89. Theorem 1. // S\, . . , S m are the different transformations of a conjugate set of a transitive linear group G in n variables, then the matrix M = S,+ ...+&, is commutative with every transformation of G and has the form of a similarity-transformation (a, a, . . , a), where a = mx(Si)/n. Proof. That M is commutative with any given trans formation T of G is seen as follows. We have T- 1 MT=T- l SiT+T- 1 S*T+ . . =Si+S 2 + . . = M, since T~ 1 S\T, . . , T~ l S m T are the transformations Si, . . , S m over again in some order ( 29, (c)). Hence MT=TM. We shall proceed to prove the second part of the theorem. By the method of 21, we can find a linear function of the variables of the group which is an invariant of M, say (xi) =axi, where a is a constant (possibly zero). 122 FINITE COLLINEATION GROUPS There may be other linear functions x z , . . such that let xi, x 2 , . . , x k be all those that have this property and are linearly independent. We shall indicate this fact symbolically by (x i} . . , x k )M = a(xi, . . , x k ). It is plain that a linear function having this property must be a linear function of the variables x\, . . , x k . Now let T be any transformation of G, and let y 2 , . . , (x k )T = y k . Then, since (zi, . . , x k )MT=(xi, . . , &) TAT, we have so that 2/1, . . have the property ascribed above to xi, . . . It follows that 2/1, . . are linear functions of xi, . . ; that is, the latter variables form an intransi tive set of G (20), unless k = n. Hence, since G is transitive, k = n, and M has the form of a similarity- transformation (a, a, . . , a). Finally, to find the value of a, we observe from the formation of M that the sum of the elements of its princi pal diagonal, ri, equals the sum of the characteristics of Si, . . , S m . But these are all equal (86, 3); hence 90. Theorem 2. Let the number of transformations in the h different conjugate sets of a transitive group G in n variables be g\, g z , . . , Qh, and let the corresponding characteristics be denoted by \\, X2, . - , Xh ( 86, 3). Then where c 8tv , . . . represent certain positive integers or zero. THE THEORY OF GROUP CHARACTERISTICS 123 Proof. Let the sum of the matrices of the g k trans formations of the kth set be denoted by M k . Then 8 (8, =1, 2, . . ,h). For the g s g t matrices in the product M 8 M t must make up one or more conjugate sets, since T~ l (M s M t )T = (T- 1 M 8 T)(T- l M t T)=M s M t . Accordingly, this prod uct is the sum of one or more of the matrices Mi, . . , M h , possibly repeated a certain number of times. We now substitute in (2) the canonical forms of the matrices Mi, . . , M h as given by Theorem 1, and obtain the equation (/3, . . , 0) = (y, . . y, ), where ft has for value the left-hand member of (1), and y the right-hand member. COROLLARY. // a transitive linear group G in n variables contains two characteristics Xs> Xt such that the sum of the n 2 roots in the product XsXt cannot be written as a sum in which primitive roots of index k are absent, then there is in G a characteristic containing roots of index k and therefore a transformation whose order is k or a multiple of k* This follows from the equation (1). By the condi tions of the corollary, at least one of the characteristics Xv of the right-hand member must contain roots of index k. There is, therefore, a transformation whose order is divisible by k. For, the order m of a transformation S = (a, /3, . . ) is the least common multiple of the indices of the roots a, /?, . . , since S m = (l, !,..) = To illustrate, let x = where i V\ and a is a primitive fifth root of unity. Here \sXt) or 4ia -\-2ia? 2a a 3 , cannot be written as a *Burnside, Theory of Groups, 2d ed., p. 347. 124 FINITE COLLINEATION GROUPS sum which is free from roots of index 20 (namely ia, ia 2 , etc.) by Kronecker s theorem ( 133, 6). EXERCISES 1. Selecting the h equations (1) obtained by keeping s fixed while taking t = l, 2, . . , h, prove that g s Xs/n is an algebraic integer (Frobenius). 2. Prove that if a group in n variables contains transformations of orders p and q, two different prime numbers both greater than n-f 1, then the group contains a transformation of order pq. 91. Theorem 3. The sum of the characteristics of a transitive group in n variables, n==^2, is zero. Proof. Let, as in 90, the group G contain h conju gate sets, and let M k represent the sum of matrices in the fcth set. By 89, the matrices M\, . . , M h all have the form of a similarity-transformation; the same will there fore be the case with the sum M of all the matrices in G, which is Mi+M 2 + . . -\-M h , say M=(t, e, . . , e). Now, if S is any transformation of G distinct from the identity, the relation MS = M ( 27, Exercise 3) is found to imply e = 0. Hence, 01x1+02x2+ . . +ghXh = Q- When w=l, two cases arise. If the group consists of E repeated g times ( 85, 2), the sum in question is g; if the transformations are not all the identity, the sum vanishes. For an illustration take the group of order 4m (each of the following transformations repeated m times): E= (1), S= , S z = (-1), S* = (-*); t-V^-1. Combining these results and referring to 86, 4, and 85, 1, we get the COROLLARY. The sum of the characteristics of an intransitive group G is its order multiplied by the integer representing the total number of the component groups THE THEORY OF GROUP CHARACTERISTICS 125 in G whose transformations are the identity repeated g times. It is plain that to each group in G of the latter type we have an absolute invariant of the first degree, and vice versa. Hence we may state the preceding corollary in the following form: the sum of the characteristics of a group G is its order multiplied by the total number of inde pendent absolute invariants of G of the first degree. EXERCISE Prove that the average number of letters which remain un changed by a substitution of a transitive substitution group G is equal to unity. (Hint: Prove that G possesses a single absolute invariant of the first degree.) ON THE CHARACTERISTICS OF ISOMORPHIC GROUPS, 92-94 92. Composition of two groups. Let G and G" be two simply isomorphic groups in respectively n and m variables, say x\, . . , x nj and y\, . . , y m . Then the nm products x\y\, . . , x n y m are transformed into linear functions of themselves when operated upon simul taneously by two corresponding transformations S and S". Hence, regarding these combined transformations as a single linear transformation S in the nm variables, we obtain a linear group G simply isomorphic with G and G". For, to E of G and G" will correspond E of G, and if SiSi = S r , S JS j = S r , we have 8^ = S r . We shall say that G is compounded from the groups G and G". LEMMA 1. The characteristic of a transformation S contained in a group G which is compounded from the groups G and G", is equal to the product of the characteristics of the corresponding transformations S and S". This is seen immediately when S and S" are both written in the canonical form, which can obviously be 126 FINITE COLLINEATION GROUPS done. If then (x a )S f = a a x a , (y b )S" = PbVb, we find = a. a p b (x a y b ), so that = 1 6 = LEMMA 2. Let there be given an invariant of G: f=X l Y l + . . +X k Y k , where X\, . . , Xk are linear functions of Xi, . . , x n and Yi, . . , Y k linear functions of y\, . . , y m . Then if k<n, the group G is intransitive. Proof. For the sake of simplicity take fc = 3, say f=XiYi+X2Y2+XtYt. We may assume that Xi, X 2 , X* (as well as Y\, F 2 , F 3 ) are linearly independent of each other; if it were possible to write, say, Xs^ai the function / could be expressed in two terms : A transformation of G will change X\, X*, X 3 into there linearly independent functions of Xi, . . , x n , and the corresponding transformations of G" will change Yi, F 2 , F 3 into three linearly independent functions of t/i, . . , y m . Let the resulting expression be and we should have /=/ . But this implies that X{, X^ X 3 are linear functions of X\, X%, X*. Hence, if n>3, G is reducible and accordingly intransitive. 93. Theorem 4. Let a transitive group G be com pounded with its conjugate-imaginary group G. The re sulting group is intransitive, and among the component groups into which it breaks up will be found just one group THE THEORY OF GROUP CHARACTERISTICS 127 made up of the identity repeated g times. Hence (Corol lary, 91), (3) 01x1x1+02x2x2+ . - +OhXkXh = g- Proof. Let the compounded group be denoted by H. Its characteristics are \tXt (Lemma 1, 92); and their sum 2 (i.e., the left-hand member of (3)), divided by g, represents the number of absolute invariants of H of the form XiXi+ . . +X n X n (Corollary, 91) ; or, as we may write it by a proper distribution of the terms, . +X n x n . We know one such invariant already, namely the Hermitian invariant ( 18), and we may assume that the variables are originally so chosen that this invariant is . . -rx n x n . Then if X is any constant, the expression /-|-X/= (Xi-{-hXi)Xi-\-(X2-\-kXz)X2-\- . . -j-(Jf n -}-X:r n ):r n is also an invariant. Now, the constant X may always be determined such that Xi+^Xi, . . , X n -\-Xx n are not linearly independent. (For an illustration, take n = 2, X\ = pxi+qxz, Here X is a solution of the equation q = 0.) r s+X Therefore either G is intransitive (Lemma 2, 92), or /+A.7 vanishes identically. Hence, since the first alternative violates the assumption of the theorem, any invariant/ of H is merely a constant multiple of / (viz., /= X/); in other words, the number 2/0 of linearly independent invariants of H is unity. The theorem follows by 91. COROLLARY. The number of variables n of a transitive linear group G is a factor of the order g. 128 FINITE COLLINEATION GROUPS Proof. Dividing the equation (3) by n we get The quantities i are algebraic integers (Exercise 1, 90), as well as the quantities \t- Hence, since the sums and products of algebraic integers are again algebraic integers, it follows that g/n is an algebraic integer; i.e., g/n is an ordinary integer ( 134). 94. Theorem 5. Let G r and G" be two simply iso- morphic transitive groups, whose corresponding character istics are x t o^nd x t> Then the sum (4) equals g or zero, according as the two groups are equivalent or not. Proof. If they are equivalent, the variables of G" may be chosen so that the groups are identical, and the conjugate-imaginary groups G and G" will be identical also. The sum (4) is then equal to g, by Theorem 4. Conversely, if the sum equals g, the two groups are equivalent, as we shall proceed to prove. Let the vari ables of G and G" be respectively Xi, . . , x n ; y\, . . , y m ; and suppose that n^m. The expression (4), divided by g, equals the number of absolute invariants of the form X\Yi+XzYz-\- . . . of the group G compounded from G and G" (91). Since there is one such, n m (Lemma 2, 92), and we may choose the variables of G" so that the invariant has the form I = xiyi+ xzy2+ . . +x n y n . But this is the form of the invariant of the group H compounded from G and its conjugate-imaginary group G , after the variables of the latter group have been THE THEORY OF GROUP CHARACTERISTICS 129 written 3/1, .., y n in place of x\, . . , x n . It is there fore an easy matter to prove that the groups G" and G are identical. For, let S , S , S" be corresponding trans formations of G , G , G", and we have (I)S S = (I)S"S t so that (I)S =(I)_S." (The transformations S and S , as well as S and S", are commutative, since the variables of the respective groups are independent of each other.) It follows directly that S and S" are identical. Finally, we cannot have more than one invariant of the form / above, since otherwise the groups G and G" would be intransitive (cf. proof of Theorem 4). Accord ingly, the sum (4) is either g or zero. COROLLARY 1 . // the corresponding characteristics of two simply isomorphic groups are equal, the groups are equivalent. COROLLARY 2. Let x t and x t be corresponding char acteristics of two groups, G and G", the first of which is transitive. The sum (4), divided by g, will in this case represent the number of components in the group G" which are equivalent to G r , when the former is broken up into its ultimate sets of intransitivity. We write G" in such an ultimate intransitive form (cf. 85, 1), and apply 86, 4, and Theorem 5. EXERCISES 1. Let H and H" be two simply isomorphic intransitive groups, and let their corresponding characteristics be denoted by x t an d x t- Then when these groups are split up into component transitive groups, the latter may not all be non-equivalent. Let us suppose that GI, G-2 , . . , G n represent types of all the non-equivalent groups obtained, there being a : and 61 of the first type in H and H" respectively, a 2 and 6 2 of the second type, etc. Now prove that the sum (4), 94, is here equal to the integer 0(ai&i+a 2 & 2 + . . +a n bn). 2. Prove that if a transitive linear group of order g in n vari ables contains a subgroup of order / composed of similarity- transformations, then g is divisible by fn (Schur). (Hint: Prove 130 FINITE COLLINEATION GROUPS first that if xt does not vanish, there will be/ distinct sets of conju gate transformations for which the products gtxtxt in (3), 93, have the same value.) 3. Prove that a group G containing a subgroup P of order p, a prime number ^n 2 /(n 1), is intransitive, unless P is invariant in a larger subgroup of G. (Hint: With reference to the sum (3), prove that the sum of the terms xtxt corresponding to the transfor mations of order p in P alone is ^npri 2 , by Exercise 1 above. Hence prove that the sum of such terms from all the subgroups conjugate to P and from the identity exceeds g.} ON THE TOTALITY OF NON-EQUIVALENT ISOMORPHIC GROUPS, 95-99 95. The regular group. We shall now consider the regular substitution group H ( 47). Regarded as a linear group, it is intransitive, since it possesses an abso lute invariant of the first degree, namely the sum of the letters of substitution. As an illustration, take the regular group on the letters Xi, . . , x 6 , simply isomorphic with the symmetric group on three letters. Its substitutions are as follows: Si = E, 4 83 = If we now introduce new variables zi, . . , 2 6 , where the transformations of H will all be of the following type: "a 0" 060000 c d e f c d L.O e f_ (5) THE THEORY OF GROUP CHARACTERISTICS 131 showing that H is split up into 4 component transitive groups in respectively 1, 1, 2, 2 variables. The groups in the variables 23, z 4 and z 5 , z 6 are equivalent, both being generated by the transformations & = (<*> 2 , *)&"" It AT Hence, there are three non-equivalent groups. In general, a regular group H on g letters, when written as a linear group in g variables, is intransitive. Cor responding to its ultimate sets of intransitivity we have a number of component transitive linear groups, which are not all non-equivalent. If we select a representative of each set of equivalent groups, we get, say, k representa tives, forming a set of non-equivalent component groups of H. Let them be denoted by H , H", . . , #<*>, and their corresponding characteristics by \t, x t, , X^- Hence, if H contains n groups equivalent to H , n" group equivalent to H", . . , the characteristic of the transformation S t of H is ( 86, 4) (6) x($)=n xl+n"x7 + - - +n<x ( f>, and this sum is =g or =0, according as S t = E or S t ^E ( 86, 5). 96. Theorem 6. Let G be a transitive linear group in n variables, and let H be a regular substitution group on g letters simply isomorphic with G. Then among the com ponent transitive groups into which H breaks up, there are just n groups equivalent to G. The theorem follows from Corollary 2, 94, if we let G and H represent G f and G" respectively. The sum (4) will here be equal to ng (cf. 95). The number k of non-equivalent component groups of H is equal to h ( 99). Anticipating this result, we can now say that the number of non-equivalent transitive linear groups to which a given transitive linear group G 132 FINITE COLLINEATION GROUPS is simply or multiply isomorphic (cf. 32, 85) is equal to the number of sets of conjugate transformations of G. EXERCISE Let the component groups H t . . , # w of H contain respec tively n t n", . . , n (A) variables. Prove that (cf. Exercise 1, 94) At least one of the numbers n , . . is unity. If there is more than one, then H possesses a relative invariant and is not simple. 97. Theorem 7. Given any two transformations S 8 and S t of H. The sum vanishes if S s is not conjugate to S^ 1 ; if these two trans- formations both belong to the conjugate set denoted by the subscript s, then 3 = g/g s . Proof. Let the equation (1), 90, be multiplied by n 2 , and a corresponding equation formed for every one of the non-equivalent groups H , . . , H w of H. The con stants c stv are the same for each of these groups, since they are the same for all simply isomorphic groups. Hence, adding the resulting k equations, the left-hand member of the new equation, say Fi = Fz, will be g s g^, and the right-hand member the expression k Now, 2^. n(w) xV is t ne sum (6), 95, for the subscript W=l v. Hence, this sum vanishes or is equal to g, according as S v does not or does represent the identity. Moreover, THE THEORY OF GROUP CHARACTERISTICS 133 in the latter case g v = l. Accordingly, the right-hand member is equal to gc^i, under the assumption that The value of c st i is found from (2), 90, and represents the number of times the transformation E occurs in the product M s M t . Now, let R be a transformation of M s . If R~ l is found in M t , then it is readily seen that M t is the sum of the inverses of the transformations of M s ; in that case E will occur g s times in the product M s M t , and g s = gt = c st \. On the other hand, if R~ l does not occur in M t , neither will the inverse of any transforma tion in M s ; in this case c st i = Q. Interpreting the equa tion Fi = F 2 under these results we finally prove the theorem. The second alternative in Theorem 7 can evidently be stated in the form (cf . 86, 2) : x/x/+x s "x/ + . . +xs (k) xs (k) = g/g a . EXERCISES 1. Prove that the set of non-equivalent groups simply iso- morphic with the alternating group on 5 letters consists of the group (H), 82; the group (Hi) obtained by interchanging the numbers Mi and /*2 in (H) ; and two groups in 4 and 5 variables respectively. 2. Prove that the group (H), 82, compounded with its conjugate- imaginary, splits up into three transitive groups, one in 1 variable (the identity repeated 60 times, corresponding to the Hermitian invariant), one in 3 variables, and one in 5 variables. 98. Group-matrix. Let x\, . . , x n be the variables of a transitive or intransitive group G=(Si, . . , S g ), and let 7/1, . . , y be g variables, independent of each other and of xi, . . , x n . Then the matrix (cf. 87) . +y g S g is called the group-matrix of the group G. Thus, the 134 FINITE COLLINEATION GROUPS group-matrix of the regular group H of order 6 given in 95 has the form (7) M = Vl 2/2 2/3 2/4 2/5 2/6 2/3 2/1 2/2 2/5 2/6 2/4 2/2 2/3 2/1 2/6 2/4 2/5 2/4 2/5 2/6 2/1 2/2 2/3 2/5 2/6 2/4 2/3 2/1 2/2 .2/6 2/4 2/5 2/2 2/3 2/1. The n 2 elements in M are linear homogeneous func tions of y\j . . , y g . These functions may not all be independent of each other; thus, .there are only 6 inde pendent elements in M of (7). Bearing in mind the definitions of 85, 2, we shall prove the following: THEOREM 8. Let H , H", . . , #<*> form a com plete set of non-equivalent, simply isomorphic, transitive linear groups in respectively n , n", . . , n w variables. Then the (n ) 2 +(n") 2 + . . + (n^) 2 = g elements con tained in the k group-matrices ?/i$i+ . . +2/A of these groups are all independent functions of the variables Proof. Consider the group-matrix M of the regular group H of which H , H", . . , H (k) are component non-equivalent groups. The matrix M will, while H has still the form of a substitution group, contain just g independent elements. For, each element will consist of a single letter y j} as may be proved easily (cf. 47). Now let H be broken up into its different component groups by means of a linear transformation T (cf. 13). Correspondingly M is transformed by T into a new matrix M : T~ l MT=T- l (y l S l + . . +y g S a )T = THE THEORY OF GROUP CHARACTERISTICS 135 The elements of M are linear functions of the elements of M, and vice versa; the coefficients being functions of the elements of T. Hence there are as many independent functions of y\, . . , y g among the elements of M as among the elements of M; namely g. Now, each non-equivalent component group H (i) is re peated as an equivalent group n (j) times ( 96), and these equivalent groups can be made identical by a proper modification of T. Correspondingly, the n (j-) matrices involved in M will have the same elements. Hence, there will be at most as many independent elements in M as are found in a set of non-equivalent groups, namely ( n )2+( n ")2_j_ _ _ +( w (*))2. But this number is equal to g (Exercise, 96). It follows that all these elements are independent, and the theorem is proved. In the case of the group H of order 6, the matrix M will have the form (5), 95, where now a = 2A + 2/2 +2/3 +2/4 +2/5 + 2/6 & = 2/i+2/2+2/3 2/4 2/5 2/6, /6> /=2/i+ a) 2/2+ w2 2/3- EXERCISES 1. Prove that the n 2 elements of each of the matrices of the transformations of a transitive group G do not satisfy a linear homogeneous equation, whose coefficients are the same for every transformation (Burnside). 2. Prove that if a certain element o u c vanishes in every trans formation S = [a a t] of a group G, the subscripts u, v being given, then G is not transitive (Maschke). 99. Theorem 9. The number k of non-equivalent transitive linear groups into which the regular group H breaks up ( 95) is equal to the total number of sets of conjugate substitutions of H. In other words, a given transitive linear group G can be simply or multiply isomorphic with just h non-equivalent transitive linear groups, including 136 FINITE COLLINEATION GROUPS itself and the group consisting solely of the transforma tion E. The proof follows that of Theorem 8 closely, after we have first made equal to each other those of the variables 2/i, . . . , y a which are factors of conjugate transforma tions in the matrix M. If therefore H contains h con jugate sets of respectively g\, . . . , QH transformations, we shall have h independent variables, say 1*1, . . . , v h . The matrix M now has the form of a transformation in canonical form. Thus, the matrix M in the group of order 6 given in 95 becomes M =(a, b, c, c, c, c), where a = v\ + 2y 2 + 8^3, b = vi + 2% 3%, c = v\ v 2 . In fact, it follows by Theorem 1, 89, that, as far as the variables of H (j) are concerned, M will appear in the form of a similarity-transformation (ft, ft, . . . , ft), where If H& and # ( > > are equivalent groups, ft=ft-/; if and # (/) are non-equivalent, ft/=t=/2/ (cf. 94, Corollary 1). Accordingly, among the g multipliers of M in its new form, there will be just k that are distinct, and these can certainly not furnish more than k expressions linearly independent in v\, . . . , v^ On the other hand, the matrix M will contain just h linearly independent elements, namely Vi, . . . , tv Hence k^h (cf. proof of Theorem 8), and h of the multipliers ft, . . . , ft are linearly independent, say ft, ft, . . . , ft. These h expressions can therefore not all vanish unless v\ = v% = . . . = *-0. However, if k>h, the expressions ft, . . . , ft must all vanish if for v\, . . . , Vh we put, respectively, the conju- gate-imaginaries of the characteristics x! A+1) > > X ( f +1) of the group G h +i- ( 94). But these quantities, THE THEORY OF GROUP CHARACTERISTICS 137 Xi h+l \ - i Xh h+l} are n t all zero ( nc f them represents the number of variables of H (h+l) ). We conclude that k>h. Hence, finally, k = h. AN APPLICATION OF THE PRECEDING THEORY 100. Theorem 10. No simple group can be of order p a q b , p and q being different prime numbers.* The proof is divided into two parts: (A). If H is the regular substitution group simply isomorphic with a group of order g = p a q b , assumed simple, then one of the component non-equivalent, transitive linear groups H , . . , H w contains q a variables, and one of the conju gate sets of H contains p? transformations. (B). Under these conditions an impossible equation is derived. (A). The relation = py = (n ) 2 +(n") 2 -h . , +(n<) 2 with the conditions that the numbers n f , . . . , n w are all factors of g ( 93, Corollary) and that only one of them is unity ( 96, Exercise), implies that at least one of them is greater than unity and prime to p; say n (t) = Again, the relation g = p a q b = g l + g 2 + . . . +& with the conditions that the numbers firi, . . . , fito are factors of g ( 29) and that only one of them is unity (or there would be an invariant operator), implies in the same manner that one of them is a power of p; say g s = (B). We now have a transitive group H w in n (t ) =^ 1 variables, and a conjugate set of g s = p? transformations. Let S (i) denote one of the transformations of this set, \ (t) its characteristic, and T the corresponding trans formation (substitution) of H. We have ( 90, Exercise 1, and 133, 7): * Burnside, Proceedings of the London Mathematical Society, Series 2, I (1904), 388-92. 138 FINITE COLLINEATION GROUPS where K represents the sum of a finite number of roots of unity. It follows that x (t) = Q a K f ) K being such a sum also (I.e.). But, x (/) is already the sum of q a roots of unity. Hence, either all these roots are alike, or x (f) = 0*. The first supposition makes S (t) a similarity-transforma tion, which would be self-conjugate in H w . This being impossible for a simple group, we infer that x ( ^=Q; and this not only for the group H (t \ but also for every one of the groups H , . . , H (h) (and their equivalent groups), the number of whose variables, like n (t \ does not contain p as a factor. Hence, the characteristic of T in H is the expression when account is taken of the fact that one of the numbers ft , . . . , n (h \ say n f , is unity, and that the correspond ing characteristic x (1) = l (cf. 95). Accordingly (86, 5), l+pK"-Q. But such an equation is impossible by Kronecker s Theorem ( 133, 6). We conclude that H is not simple. EXERCISE Prove that a group in which the number of operators in a con jugate set is the power of a prime number, is not simple (Burnside). *This follows directly from 133, 6. CHAPTER VII THE LINEAR GROUPS IN FOUR VARIABLES 101. Introduction. We shall again adopt all the conventions laid down in 51, and we shall employ the same classification for the groups now under discussion as that used for the groups in three variables ( 75). How ever, we shall begin with the primitive simple groups, the construction of which is the most difficult problem in the present chapter; and, proceeding at a comparatively slower pace while dealing with these groups, we shall determine completely the generating transformations of every type under this head. On the other hand, there are a host of types of the somewhat more easily constructed groups in four variables which are non-primitive or contain non-primitive invari ant subgroups. We shall therefore not attempt to list all of these groups, but shall give an outline of the theory and so much of the detail that the student may encounter no serious trouble in constructing such of these groups as may be needed for any purpose. The following propositions are of constant use and may be proved by the student (cf. 31, Exercise 3): 1. A set of conjugate operators of a group G generate an invariant subgroup of G. 2. If the generators of G all possess an invariant configuration (function, equation, point, line, plane, etc.), then G will possess the same invariant configuration. For example, to examine if G is monomial, it is sufficient to try out the generators of G for an invariant set of four planes XiXzXzX^Q. (Cf. 77, where the solution of this problem is indicated for a group in three variables.) 139 140 FINITE COLLINEATION GROUPS Again, to examine if G is intransitive or imprimitive in two sets of two variables each (cf. 120), we write down two sets of two linear functions of the variables with undetermined coefficients : The conditions that each generator of G will transform these sets as sets of intransitivity or imprimitivity will furnish a number of equations among the 16 coefficients a\ . . that must be simultane ously fulfilled. Thus, if a generator A permutes the two sets, the expressions in the first set are by A transformed into linear functions of the expressions in the second set, and vice versa; that is, certain constants \, . . , ^4 can be found such that the following identities in xi, x 2 , x 3 , x* are fulfilled: It is apparent that the labor involved in a problem of this kind is somewhat tedious, but is not difficult. Notation. Throughout this chapter the letters i and o> represent the primitive fourth and third roots of unity, i = l/^l, a>=(-l+iV3)/2, respectively. Multipliers. We shall often have occasion to refer to the multipliers or characteristic roots of a transforma tion ( 2, 23). These multipliers are in such cases inclosed in brackets: [a, ft, y, 8]. Thus, the multipliers of the transformations A\, A 2 , A 3 , A 4 of (9), 123, are all as follows: [1, 1, -1, -1]. We call to mind that conjugate transformations have the same multipliers ( 23) ; hence if a group G contains a Sylow subgroup of order 7 generated by a transformation whose multipliers are [1, jS, 4 , /3 2 ], the multipliers of any transformation in G of order 7 are either [1, /?, 4 , ft 2 ] or [1, 3 , ft 5 , ft 6 ] ( 36). Type. As hitherto, any one of a series of equivalent groups (51, 4) may be selected as a type of the groups of the series. One group may thus have the type of another group without being written in the same form. For instance, the group (A), 102, has 10 subgroups all of type (c), 111, though only one of them has the LINEAR GROUPS IN FOUR VARIABLES 141 canonical form. It also possesses 6 subgroups of type (g), none of which have the monomial form in the variables chosen for (A). There are in all 30 types of primitive groups in four variables. These types are here designated as follows: (A) (F),102; (G) (K),119; l-7, 121; 8-12, 122; and 13-21, 124. THE PRIMITIVE SIMPLE GROUPS, 102-117 102. List of the groups. The groups (A)-(D) are isomorphic with the alternating substitution groups on 5-7 letters: abcdefg. The first three are constructed by means of Moore s theorem (50; cf. 58, 82), which gives two solutions for the group of order 60<.* The groups (D)-(F) are obtained in 106-109, 115, 116. (A) Group of order 60< generated by F 1} F 2 , F s , where Fi- (1,1, *,*); The corresponding substitutions of the alternating group are (abc), (ab)(cd), (ab)(de). (B) Group of order 60 generated by FI, Fi, F 3 , where *Seo Maschko, Mathematische Annalen, LI (1899), 278-89. A number of the types given by Maschke are intransitive. To obtain the group (D) by Moore s theorem wo should have to start with one of these intransitive groups. 142 FINITE COLLINEATION GROUPS (C) Group of order 360< generated by FI, F 2 , F 3 of (A), and F 4 : Xi = xi, x 2 = x[, z 3 = -x 4 , z 4 = -x 3 . The corresponding substitutions of the alternating group are (abc), (ab)(cd), (ab)(de), (ab)(ef). (D)* Group of order J 7 ty = 2520< generated by S, T, W, where ft- ft A Aft i T: Xi = x(, x z = Xs, x 3 = Xt, X4 = xi; W: Zi = m(p 2 4+^+Z3+4), X 2 = m(x l -qx 2 -px f ,-px 4 ) ) where The corresponding substitutions of the alternating group are (abcdefg), (bce)(dgf), (bce)(dfg). (E)f Group of order 168<#> generated by S, T of (D), and R : zi = n where n = 77s, v i (F)J Group of order 2 6 -3 4 -.5< = 25920< generated by T of (D) and C, D, 7, F, where C = .(1,1,0,,^); D=Ku,,u>, 1); F: x^xl, 5Ct-Jb(a4-ha4--aO, z 3 = /K^+" F: Xi=-4, aj2 = 4, 2:3=-^, x 4 =-^. * Cf. Klein, Mathematische Annalen, XXVIII (1887), 519; Maschko bid., LI (1899), p. 291. f Cf. Maschke, Papers of the International Mathematical Congress, Chicago, 1896, p. 176. t Jordan, Traite des Substitutions, Paris, 1870. 318; Maschke, Mathe matische Annalen, XXXIII (1889), 320. LINEAR GROUPS IN FOUR VARIABLES 143 103. Geometrical properties of transformations of variety 2. There are two kinds of transformations of variety 2, according to the way in which the multipliers are repeated: [a, a, /?, /?] and [a, a, a, /?]. 1. Interpreting x\ t x 2 , x 3 , x 4 as homogeneous co-, ordinates of space, a transformation of the first kind, written in canonical form $=(<*, a, /?, /?), will leave invariant every straight line of the family axi+bx 2 = Q, cx s -\-dx4 = Q, where a, b, c, d are arbitrary constants. Correspondingly, there is a family of invariant lines associated with any other transformation T having the same multipliers as S; and since at least one line can always be found belonging to two such families, it follows that S and T have an invariant line in common. If the variables are changed so that this line is x\ = 0, x 2 = 0, the group H generated by S and T is reducible (xi and x 2 being transformed into linear functions of themselves by both S and T). Hence, if S and T belong to a finite group G, the group H is intransitive ( 20). Let its sets of intransitivity be (x\ t x 2 ) and (x 3 , z 4 ), and its operators will have the form Ci, 14. Then if one of the transfor mations is written in canonical form as far as the variables of any one set are concerned, the corresponding multipliers may be either [a, a], [/?, /?], or [a, /?]. In the first two cases S and T are commutative ( 6, Exercise 2). Accord ingly, if they are known not to be commutative, their multipliers must be [a, /?] for both sets of intransitivity. 2. The transformation A = (a, a, a, /?) leaves invariant every plane through the point x\ = 0, x z = 0, 3 = (namely, every plane whose equation is pxi+ 5x2+^3 = 0). If B is a transformation having the same multipliers as A, and therefore possessing a similar invariant configuration, the group H generated by A and B must leave invariant every plane through a certain line, namely, the line which joins the two invariant points in question. Moreover, if 144 FINITE COLLINEATION GROUPS A and B belong to a finite group G, and if the variables are so changed that the invariant line is x\ = 0, x 2 = 0, the group H is intransitive, its sets of intransitivity being (zi), fe), and say (z 3 , z 4 ). As in 1, the multipliers of A or B corresponding to the intransitive set (z 3 , z 4 ) are [a, j8], unless A and B are commutative. Now assume that A and B are of order 5, so that a and ft are 5th roots of unity. The component of H in the variables (z 3 , 4 ), which we shall indicate by H , must then be reducible to the group (E), 58, by the process of 12. This group can be generated by the operators of order 2$ = 4 that it contains ; the corresponding operators of H must have the multiplier [1] in the sets (xj) and (z 2 ) and will consequently have the multipliers [i, i] in the set (#3, #4) (cf. 51, 3). These operators are accordingly of determinant unity in H ; and the group generated by them, namely (E), is therefore a subgroup of H . Hence, finally, H contains a transformation of order 3<, whose multipliers are [ , w2 ], and correspondingly H contains a transformation whose multipliers are [1, 1, to, o> 2 ] and is not primitive ( 70). Next assume A and B two non-commutative conjugate operators under G, of any order. The line Xi = x 2 = invariant under A and B cannot be invariant under every conjugate to A, unless G possesses an invariant intransi tive subgroup ( 101, 1). Hence assume a conjugate C which does not leave this line invariant. There is, how ever, an invariant plane common to A, B } C, which passes through the line Zi = 2 = 0, as is readily seen; and we can so choose the variables that this plane is x\ = 0. The group K generated by A, B, and C is accordingly in transitive, its sets of intransitivity being (xi) and (x z , x 3 , z 4 ). Moreover, the component of K in the vari ables (x 2 , Xzj z 4 ) is not intransitive by virtue of the assumed facts that A and B generate a transitive group LINEAR GROUPS IN FOUR VARIABLES 145 in (x 3 , 4) and that C does not leave invariant the plane #2 = (or it would leave invariant the line Xi = x 2 = Q). EXERCISES 1. Prove that if G contains two non-commutative transforma tions of order 5, both having the multipliers [a, a, , /3], and if, furthermore, the intransitive group generated (cf. 1 above) is abelian in one of the sets of intransitivity and not in the other, then G contains a transformation whose multipliers are [1, 1, w, w 2 ] and is not primitive. 2. If two transformations D and A, whose multipliers are respectively K w, w, 1] and [a lt a it 03, a 3 ], where af = a-j=a| = l, belong to a finite group G, the subgroup generated by them is either abelian or is intransitive in (1, 3) variables. Moreover, the component group in the intransitive set containing 3 variables is again intransitive, since no finite transitive group in 3 variables contains two non-commutative operators of orders 5 and 3 respec tively, the latter having the multipliers [w, w, 1]. Hence prove that either the subgroup of G generated by all the conjugates to A is intransitive (all being commutative with D), or that G contains a transformation whose multipliers are [1, 1, w, w 2 ] and is not primitive. 104. Theorem 1. No primitive simple group G can contain a transformation of order 5 and variety 2. Since G can have no invariant subgroup other than itself, a conjugate set of operators of order 5 must gener ate G ( 101, 1). The theorem now follows from 103, unless every pair of non-commutative conjugate trans formations of order 5 and variety 2, having the multi pliers [a, a, fi, /?], generate an intransitive group H in two sets (xi, z 2 ), Oa, z 4 ), primitive in both sets. Consider two such non-commutative transformations, S and T. The two component groups in two variables generated by S and T must each be reducible to the group (E), 58, by the process of 12; and it is found that these groups are equivalent, since the generating transformations S and T have the same multipliers [a, (3] 146 FINITE COLLINEATION GROUPS in both sets. An appropriate choice of variables will therefore cause the corresponding matrices to be identical, so that the transformations of H all have the form a b c d a b c d (1) It is now easily proved that each line of the family where A. and /* are arbitrary constants, is invariant under H. A similar set of lines will evidently be invariant under the group H generated by S and a third transforma tion U having the same multipliers, unless S and U are commutative. Assuming that the variables of G are selected so that S has the canonical form (a, ft, a, (3) } we find that after a slight reduction the second set of lines can be defined by the equations where a, 6, c, d are certain constants depending on U, and A/, p are arbitrary constants. The families (2) and (3) are now seen to have at least one line in common, namely, one for which A = A , n = n r , (\ a+n c)n=(\ b+[ji f d)\. It follows that the group generated by H and H is reducible and therefore intransi tive. Accordingly, any two transformations T and C7, both conjugate to S and non-commutative with it, generate with S an intransitive group in two sets of two variables each. This group can be none other than H; hence, all the conjugates to S fall into two classes: those which belong to //, and those which are commutative with S. Any transformation in the latter class must be commutative with every transformation of H, since it LINEAR GROUPS IN FOUR VARIABLES 147 must have the form of a similarity-transformation for each of the two sets of intransitivity : (a, a) in one and (/?, /?) in the other. The group generated by S and all its conjugates is therefore either abelian or it contains the intransitive, group H as an invariant subgroup. In any event, G is not a primitive simple group. 105. Theorem 2. // a Sylow subgroup P of a simple group G is abelian and contains, aside from the identity, operators of order p only, where p is a prime number, then P is invariant under a larger subgroup Q of G of such a nature that not one of the operators of P except the identity is in variant under Q. This theorem is a particular case of a theorem given by the author in Transactions of the American Mathe matical Society, XI (1910), 2. The proof is long and will not be reproduced here; an outline of the principles involved is given below ( 107). 106. Theorem 3. The group (F), 102, is the only primitive simple group which contains a transformation of order 3 whose multipliers are [I, <*>, o>, w].* Proof. In a group G, let *Si, Sz, . . , Sh be a set of conjugate transformations which have the multipliers [1, w, tu, w] and which generate an invariant subgroup H. Since G is to be a simple group, we have H = G, and H cannot be intransitive by assumption. Hence ( 103, 2), we can find three transformations among Si, . . . which generate an intransitive group in sets of (1, 3) variables, say (x\), (xzj Xsj x 4 ), while the component group in the variables (z 2 , 3, 4) is transitive as far as these variables are concerned. Moreover, this component cannot be * Bagnera, Rendiconti del Circulo matematico di Palermo, XIX (1905), 19 fl. 148 FINITE COLLINEATION GROUPS imprimitive since it is generated by transformations hav ing the multipliers [o>, w, 1] (cf. 86, 5). Now, consulting the groups in three variables (chap, v), we discover that only one primitive group possesses operators of order 3< and variety 2, namely, the group (G), 79, of order 216<. This group may be written as a group K in four variables so that the operators just mentioned (namely, those conjugate to U, 79) become transformations having the multipliers [w 2 , w 2 , w 2 , 1] in K. The similarity-transformation (w, w, w) in (G) becomes the transformation (1, w, w, o>) in K; the latter group is therefore of order 216-3 = 648. 107. A Sylow subgroup P of order 81 is generated by the transformations in K corresponding to Si, T, U ( 77, 79). It contains an invariant abelian subgroup PI of order 27, generated by Si=(l, 1, o>, <o 2 ), T^SiT^ (1, o> 2 , 1, <o), and 7=(o> 2 , o> 2 , o> 2 , 1); and it follows from the general theorem referred to in 105 that G must contain a transformation W\ which transforms PI into itself and which transforms Xi into a variable different from x\. The successive steps in the proof of the the orem as applied to the case under consideration are as follows : 1. The operators of G transform the plane Xi = into a number of distinct planes, say x\ = 0, x{ = 0, x" = 0, . . . The geometrical configuration composed of these planes: . . =0 is transformed into itself by G; i.e., J is an invariant of G. Moreover, since G is simple, J must be an absolute invari ant ( 88), since otherwise G is isomorphic with an abelian group in one variable, namely, the variable J. 2. Now let the planes (factors) in J be separated into sets such that planes of any one set are obtained one from another by operating by the transformations of PI. The LINEAR GROUPS IN FOUR VARIABLES 149 number of planes in a set is a factor of 27, and the sets do not overlap. If TT is a subproduct of J, made up of the factors of a set, then TT is an invariant of PI. From this we find that if * contains only one factor, it is one of the variables x\ x 2 , x 3 , x*. If it contains 3, 9, or 27 factors, then (TT)U = TT. 3. The plane #i = constitutes a set by itself, and (xi) U = < 2 Xi. But, since J is an absolute invariant under G, we must have ( J) U = J. It follows that there is at least one subproduct * different from x\ and such that (TT)U = CTT, where c?^l. By 2 this factor TT is one of the variables x z , x s , z 4 . Assuming that W was the trans formation of G which transformed x\ into TT, we find that the group WPiW~ l leaves Xi = invariant (viz., (a?i)TFPiTr- 1 ()PiTr- l -(c )TF- 1 =c / a;i) and will there fore generate with K an intransitive group in (1, 3) vari ables. However, since (G), 79, cannot be a subgroup of a larger group in three variables, except one obtained by adding new similarity-transformations, it follows that WP\W~ l is a subgroup of K. 4. By means of Theorem 7, (6), 36, we can now find a transformation L in K which transforms the Sylow subgroup to which PI belongs into the Sylow subgroup to which WP\W~ 1 belongs. Since a given Sylow subgroup of order 81 contains a single abelian subgroup of order 27, it follows that L~ l P l L = WPiW~ 1 . The transformation LW is now found to possess the properties demanded of W i above. 108. The conditions W^ l PiWi = Pi, (xi)Wi = *, imply that Wi has the monomial form and transforms x\ into one of the variables z 2 , 3, z 4 , multiplied by a constant. In K we have already the monomial transformations T and F 2 which permute the variables as do the substitu tions (xi)(x 2 x s x 4 ) and (xi) (x 2 ) (x 3 x 4 ) respectively (41). It is therefore possible to multiply W\ by such a 150 FINITE COLLINEATION GROUPS transformation in the group generated by T and V 2 that this product has the form F x\ = ttx i } #2 6#2, Xz = cx\, x\ = dx[. Now, an odd power of F has the same monomial form; we may therefore replace this transformation by such an odd power of itself that the new F is of order 2". Furthermore, since F 2 = (ac, 6 2 , ac, d 2 ) and is now by assumption of order 2 n ~ l , and since no transformation in canonical form (6 2 , ac, d 2 ) belongs to (G) except such as are of order 3<, it follows that ac = b 2 = d 2 . In addition, the determinant of F is acbd= 1. Consequently (4) 6 4 ==Fl, d ===&. To determine the coefficients more fully we construct the pr6duct FV. Its characteristic is (6+do>)/i/ 3 = 6(1 o>)/3 or = &w 2 , according as d = b or = b. But 6(1 <i>)/3 cannot be written as a sum of four roots of unity (86; 133, 6) for the values of b satisfying (4). Hence we have d = b; and we may now multiply F by such a power of (z, z, z, i) that we obtain 6=1, d= 1, ac=l. Finally, the change of variables indicated by the transformation ( a, 1, 1, 1) (cf. 13) does not alter the group K, while it gives to F the form as listed in the group (F), 102. Here we have written C and D for Si and U 2 . 109. The group (F) is evidently primitive (cf. 101, 2). Concerning the statement that it is a simple group we merely observe that it is not obtained by enlarging any of the other groups in this chapter. There remains to prove that the transformations T, C, D, F, F actually generate a group of order 25920<. A group of this order cannot be a subgroup of a larger primitive simple group ( 111); hence no further generators can be added to the list (F). LINEAR GROUPS IN FOUR VARIABLES 151 The 40 planes xi = 0, x 2 = 0, z 3 = 0, x 4 = 0; (5) xi-0ix 2 +0 2 x 4 =o, si xi- 01*4+02*3 = 0; (0i, 02 = 1, >, " 2 ), are permuted among themselves by each of the generating transformations given, and therefore by (F) itself. This group is accordingly isomorphic with a (transitive) substitution group on 40 letters, and the order of (F) will be 40A: ,. where k represents the order of that subgroup K of (F) which leaves Xi = invariant ( 45). We now write down a matrix with arbitrary elements M=[a st ] to represent a transformation in the group K . The con ditions (xi)M = mxi, and that M is unitary ( 19) due to the fact that the Hermitian invariant of (F) can be none other than 0:1X1+0:2X2+0:3X3+0:4X4, will give us the form <7 2 , 14, for M . We finally impose the condition that M permutes among themselves the planes (5) ; this prob lem can be simplified by multiplying M by suitable trans formations of K.. For instance, the case (x 2 )M = ra x 3 is reduced to the case (x 2 )M = ra x 2 by substituting MT~ l for M at the outset. There results that M belongs to K, so that K = K, and the order of (F) is 25920<. 110. Theorem 4. No primitive simple group can con tain a subgroup "H p " ( 66). Let G be a group which is shown, by application of the theorems of 66-68, to contain an invariant subgroup "H p ". Then since G is here assumed simple, we must If p = 3, it follows from 67 that a transformation T whose order k is prime to 3 must have the multipliers [1, a, a, a]; a*=l. But the determinant of this trans formation, a 3 , cannot be unity. Accordingly, a group H z can contain no operator whose order is prime to 3, and is 152 FINITE COLLINEATION GROUPS therefore not primitive ( 61). Similarly, a group H p is not primitive if p>3. Consider finally a group H 2 . By 67, a transforma tion in this group of odd order q must have the multi pliers [a, a, ft fS\. Therefore, by 73, q<6. Again, by Theorem 1, q<5. The order of H 2 is therefore limited to the numbers 2 a -3 6 , and the group is not simple ( 100). 111. The Sylow subgroups. The theorems of chap, iv and the Theorems 1-4 above enable us to construct, largely by trial, a Sylow subgroup P of order p a <f> con tained in a primitive simple group G. Thus, to con struct a group of order 2 a < we make use of the facts that it can have no transformation of order 8< and variety 4 or 3 ( 66), and none of order 4< and variety 2. More generally, by following the principles of 66, 68 we may prove that a group in which are present the two commuta tive transformations (i y i, 1, 1), (i, 1, i, 1), either as here written or multiplied by similarity-transformations, must contain an invariant group "// 2 ". By trial we now find that an abelian group of order 2 4 < will always contain the invariant group "#2"; hence (74), a^ 3+3 = 6. When p>3 the group P is abelian, and Theorem 2, 105, can be applied. For instance, let us assume a group P of order 7, generated by S=(l, ft ft /? 5 ); fP=l. By Theorem 2 there must be an operator T which transforms S into a power of itself: T~ l ST = S k , where k^l. But no matrix T of non-vanishing determinant exists satisfying the equation ST=TS k . In this manner the various generators of order 7 are excluded except the following: (1, ft, P, W, 08, P, P, /3 5 ), (ft 3 , P 3 , P, W, (I, 1, A F). But the last two are eliminated by Theorem 8, 70. Similarly, every abelian group of order 7 2 or 7 3 is elimi nated. LINEAR GROUPS IN FOUR VARIABLES 153 The group Q is generally determined by Theorem 2 and 68. Thus, the cube of the operator T which transforms S= (1, ft, ft 4 , ft 2 ) into a power of itself (say $ 2 ) must be a similarity-transformation. For otherwise we should have a subgroup "H p ," since T z is commutative with S. The non-vanishing elements in the matrix of T can now all be made unity by a fitting change of variables and by multi plying T by a power of (i, i, i, i) . In listing the results we use the following abbrevia tions : 3>, the group of similarity-transformations contained inG; P, the Sylow subgroup of order p a ; Q, that subgroup of G which contains P invariantly, when p>3; T, the transformation x\ = x{, Xz 3, #3 = x\, x\ = x^ ; R, the transformation x\ = 3, x^ x ^ Xs = x!>, X4=x(; Ri, the transformation Xi = ax{, x 2 = bx^ 3 = czJ, Xi^dx s, where a, b, c, d are certain constants; V, an operator which permutes among themselves the variables 3, x*, and transforms Xi, Xz into linear functions of themselves; Qi, the group of order m<f> generated by all the trans formations of G which are not of order 5k and which are commutative with the transformation A i under (f). The letters a and ft represent respectively primitive 5th and 7th roots of unity; y, 8, (y/^S), roots of index 11; and c, , (e?^)> roots of index 13. P: Q: Group Order Generators Order Generators (a) 2*4>, (0^6); ( b ) 3, W =K<o,<o2,u>*); (C) 3, F, =(1,1, cu, o>2); 154 FINITE COLLINEATION GROUPS P: Q: Group Order Generators Order Generators (d) 9, F l9 W =(,<*, 1, 1); (e) 81, W, T(106); (f) 5, A, =(1, 1, , a*); 10m<, P, *, V, Q, , (g) 5, A, =(a,a3); 1Qm ^ p ? ^ R 2. (h) 5, A 2 ; 20<, P, <*>, #; (j) 25, A!, A 8 =(l,a, l,a<); 15*, P, *, T 7 ; (k) 25, Ai, A 3 ; 30*, P, *, T 7 , fl i; (1) 7, S = (1,/3,0*,#0; 21*, P, *, !T; (m) 7, = (ft/3,/? 2 ,/F); 14<#>, P, $, # 2 ; (n) 11, (7 =(y,y 10 ,8,8io ); 2 2</>, P, *, iP; (o) 13, D! = (c, e ^ (;i2) . 26<#>, P, *, ,R 2 ; (p) 13, D 2 =(1, c, e 3 j 9 ); 39^ P, $, T; (q) 13, D 8 -(v^^O; 52cA, P, <*>, R. It is easy to prove that no two groups of the types (e), (f), (j), (k), (1), (m), (n), (o), (p), (q) can be subgroups of a given group G at the same time. First, no primitive simple group can contain a transformation whose multipliers are [w, w, w, 1] and at the same time one whose multipliers are those of Aij by Exercise 2, 103. Thus the type (e) excludes the types (f), (j), and (k). Secondly, no primitive group can contain a Sylow subgroup of order q and types (l)-(q) and at the same time one of order p or p 2 (p^q) and types (e), (f), (j)-(q). For then we should have a transformation U of order pq ( 90, Cor.), and therefore a transformation of order p (viz., U q ) commutative with one of order q (viz., U p ). But two such transformations would imply a subgroup "H p ," p^3 ( 68). Hence, the order of a primitive simple group in four variables is g<j>, where g is a factor of one of the numbers 2 8 -3 4 -5, 2 6 -3 2 -5 2 , 2 6 -3 2 -5-7, 2 6 -3 2 -5-l.l, or 2 6 -3 2 -5-.13: 112. Reduction in the number of types of the Sylow subgroups. The types (f), (j), (k), (m), (n), (o), (p), (q) LINEAR GROUPS IN FOUR VARIABLES 155 can be eliminated chiefly by aid of Theorem 4, 93, from which theorem it follows that if the sum of the products xx exceeds the order #</> of the group G, then G is intransitive. (For an illustration, the term xx corresponding to the transformation Z) 2 is (1+e+eH-* 9 ) (l+e l2 + 10 + 4 ) =3.) Consider the type (f). In the group Q there are 5w< m<f> = 4m< transformations whose orders are mul tiples of 5. Two such transformations belonging to two different subgroups conjugate to Q cannot be identical, since the corresponding transformations of order 5 are distinct. The sum M = ^\x for the transformations of order 5fc in G will therefore be qh, where q represents the sum of such terms from Q, and h the number of sub groups conjugate to Q, namely, </</( 10ra<), by 30, Theorem 2 . The group Q is intransitive, in (2, 2) variables: (xi, 2 ), (#3, # 4 ); or in (1, 1, 2) variables. Going over the various possibilities in detail we find that the sum qis at least 10ra</> (cf. 94, Exercise 1). The sum M is therefore at least (g / (Wm)) (Wm<j>) =#<. But this number, together with the product xx corresponding to the identity, namely, 16, exceeds g<j>. In this way the cases (n) and (o) can be disposed of. 113. Consider next the type (m). By 111 the order of G is in this case a factor of 2 6 3 2 - 5- ?<#>. On the supposition that the order is divisible by 5, the group would contain 0</(10</>) or #</(20<) subgroups of order 5 (type (g) or (h)), and </</( 14<) subgroups of order 7. But these numbers should be of the forms l+5fci and l+7fc 2 respectively (36). By trial we find this impos sible except for the order 210<. But this number does not correspond to a simple group ( 48). Hence, the order of G is not divisible by 5; and therefore no transitive group isomorphic with G can be constructed in 5 variables ( 93, Cor.), a fact to be used presently. 156 FINITE COLLINEATION GROUPS Now let G be a group equivalent to (7, differing from the latter merely in being written in the variables yi, . . , 2/4 instead of i, . . , z 4 . The six functions (6) Vn are readily found to be transformed into linear functions of themselves by the intransitive group made up of G and G as component groups. We can therefore regard these six functions as the variables of a group H simply iso- morphic with G and G . The transformation B becomes (1, 1, ft, /8 6 , /3 3 , /3 4 ), and R 2 will permute the variables as indicated in the cycles (^12) (^34) (^23^14X^24^13). By adding the terms xx corresponding to the transformations of order 7< we find that H is intransitive. Hence, since no transitive component can contain 5 variables by what has been proved above, and since no simple group in 2 or 3 variables contains an operator of order 7 which is trans formed into its inverse by one of order 2 (viz., R 2 ), it follows that the components of H are three in number, embracing the variables (#12), (^34), (^23, , ^13). In other words, the two functions Vi 2 = Xiy^x^yi, ^34 = ^32/4 z 4 2/3, are invariants of H. But it is easily proved that a transformation in four variables whose corresponding operator of H transforms each of these functions into a constant multiple of itself must have the form Ci, 14. The group G is therefore intransitive. In the case (q), the group H is first proved to be intransitive in two sets (viz, #34), (^23, . , ^13), with R permuting the vari ables in the first set, and with D 3 represented by the iden tity (1, 1) in this set. But no group in two variables (^12, VM), transitive or intransitive, and of order 2n, can be isomorphic with a simple group of order 52kn (cf. 32). In the case (p), the group H is likewise intransitive, the sets now embracing (3, 3) variables: (^12, ^13, ^14), LINEAR GROUPS IN FOUR VARIABLES 157 fes, ^34, *>4i). But no simple group in three variables is of order 13fc (cf. chap. v). 114. Finally, assume that the group G contains a group P of type (j) or (k). The order of G is then a factor of 2 6 -3 2 -5 2 -> (111). Two different subgroups of order 25 cannot have a transformation of order 5 in common. For, let A be a possible common transformation; then, since both groups are abelian, the group generated by them contains A as an invariant transformation and is therefore intransitive. If we now chose for A in turn each of the transformations that belong to P, we find in every case that the sets of intransitivity involve (2, 1, 1) variables. The component group in two variables must be derivable from the group (E), 58, and G would contain a transformation whose multipliers are [ w, w 2 , 1, 1] and is not primitive (cf. 103, 2). The total number of Sylow subgroups of order 25 is therefore of the form 1+25& (38, Exercise 2). Hence (l+25/b)-15< or (l+25fc)-30< should be a factor of the order of G ( 30, Theorem 2 ). But this is found to be impossible. Accordingly, there remain only the types (a), (b), (c), (d), (e), (g), (h), and (1) as possible Sylow subgroups of a primitive simple group G in four variables. 115. The simple group of order 7-9- k. Here we have a group Q of type (1), generated by S, <$, and T. The last transformation has the multipliers [1, 1, <o, o> 2 ] (cf. 86, 5), and belongs to a Sylow subgroup of order 9 and type (d). There is therefore in G a transformation W of order 3 commutative with T, whose multipliers are [w, w 2 , 1, 1]. Adopting such variables for G that Q has the form as given under (1), we select a matrix with arbitrary elements to represent W and impose the 158 FINITE COLLINEATION GROUPS conditions that WT =TW, and that the characteristics of W, WT, WT 2 are 1, -2, -2 respectively. These condi tions determine a matrix with only three arbitrary ele ments a, b, s: ~-2-3s a a a b s+l s s b s s+l s b s s s+l_ The characteristics of SW and S*W are [SW]= -2-3s-f (s+l)p, [&W]=-2-38+( S +l)q, where p = P+P+(P, g = 3_j_^5_j_^6. and they are each the sum of the four roots of unity which are the multipliers of the respective transformations. If one of these multipliers is a 7fch root of unity or is (more generally) of index 7n, then the four multipliers are the same as those of S or a power of S, possibly multiplied by a similarity-transformation, since we would otherwise have a subgroup "Hp" (cf. argument at the end of 111); that is, the corresponding character istic is r(l+p) or r(l+q), where r==*=l or =*=i. Kron- ecker s theorem ( 133, 6) can now be applied directly to the equation obtained by eliminating s, and we find the following alternatives : I: [SW]=-(l+q),[&W]=-l; II: [SW] = -l,[&W]=-(l+p). Selecting the first, we derive s= (4+p)/7. Again, from W 3 = E we now get a&= 1/7; and the change of variables expressed by the transformation (V a/6, 1, 1, 1) will finally give us the form of W as written in (D), 102. The second alternative (II) above would furnish W 2 instead of W. There remains for us to answer the following questions : Do the transformations S, T, W generate a group of LINEAR GROUPS IN FOUR VARIABLES 159 order 7!</2, isomorphic with the alternating group on 7 letters ? Can no new generators be added to S, T, W? Concerning the first question we remark that if a group G of order 7!</2, isomorphic with the alternating group on 7 letters, can be constructed as a linear group in four variables by Moore s theorem ( 50), then the group generated by S, T, W must be equivalent to G f . For three such operators or their equivalents are evidently present in G ; moreover, the alternating group on 7 letters contains no subgroup of order 63fc except itself. That the group G exists has been shown by Maschke (cf. footnote to 102). To answer the second question we observe that an assumed primitive simple group generated by S, T, W and additional generators must contain (D) as a subgroup by what precedes, and hence be of order 2 3+c 9 5 7< (111); c^3. Now counting the Sylow subgroups of order 5 and type (h), such groups being already present in (D), and of order 7, we may readily prove that c = is the only solution (cf. 113). 116. The simple group of order 7-3 -k. As in 115, we chose such variables for the group now under dis cussion (G) that the subgroup Q of order 21 will appear in the form (1), 111. In G there is a Sylow subgroup of order 3 generated by T, and the order of G is a factor of 2 6 -3-5-7-<. The number of subgroups of order 7 must be either 8 or 64, and correspondingly the order of G is either 168< or 1344$. Only the former number is the order of a simple group (48). This simple group can be written as a substitution group K on 8 letters abcdefgh representing its 8 subgroups of order 21 ( 46, Cor.). If a represents the group Q, and if S transforms the group b into c, etc., the operators S 160 FINITE COLLINEATION GROUPS and T are in K represented by the substitutions (bcdefgh) and (cdf)(ehg) respectively. An additional generator of G is obtained by constructing an operator R of order 2</>, which transforms T into T 2 ( 105) and whose substitu tion in K is (a&) (ch) (de) (fg) . The conditions R~ 1 TR = T 2 , R 2 = a similarity-transformation, give R = r v v v w s t u w t u . W U S t We may assume v = r; this is equivalent to changing the variables in G by means of the transformation (v/r, 1, 1, 1). To further specialize the elements in R, we note that with each subgroup of order 21 belongs an invariant plane; in the case of a this is i = 0, the remaining planes being (xi)R = Q, (xi)RS = Q, . . , (x 1 )RS Q = 0. The planes be longing to c and h are (xi)RS=r(xi-\-^x 2 +^x 3 -\-l3 2 X4)=0 and (xi)RS G =r(xi-\-(3 6 X2+P*X3-\-(3 b X4)=Q , hence, since c is transformed into h by R, we have correspondingly where k is an undetermined constant. This condi tion will fix definitely the ratios of the elements in R; adding the condition that the determinant of R is unity we finally obtain this generator as listed under (E), 102. Having proved that no operator except a similarity- transformation can leave invariant each of the 8 invari ant planes, we may show that S, T, R generate a group of order 168</>, isomorphic with the simple group of order 168 (cf. 109). * If c =0, the group generated by S, T, R is intransitive. LINEAR GROUPS IN FOUR VARIABLES 161 117. The simple groups of order 5k. We come now to the problem of determining the primitive simple groups whose orders are factors of 2 6 3 2 -5-<, and in which there are subgroups Q of order 10< or 20< as listed under (g) or (h), 111. Counting the Sylow subgroups of order 5, we discover that the orders of the groups sought are all <1000. Hence, these groups are isomorphic with the alternating groups on 5 and 6 letters ( 48), and can be constructed by Moore s theorem ( 50). There result the types (A), (B), (C) ( 102). 118. Groups which contain primitive simple groups as invariant subgroups. In the construction of such groups we are aided materially by the following theorems. THEOREM 5. // a given primitive group G does not contain a subgroup "H p " ( 66-68), neither does a larger group K in which G is contained as an invariant subgroup. 1. We shall prove that % if K contains an invariant subgroup "H p ," then G must contain such a group also. Let H be a subgroup "H p " of K, and let us assume to begin with that T is an operator common to both G and H and is not a similarity-transformation. If then V belongs to G, the transformation VT belongs to G, and the equa tion (9), 66, is true, since it is true when V is any trans formation of K and T of H. But if (9) is true, the group G contains an invariant subgroup "H p " (by the argu ments of 66), contrary to assumption. Hence, G and H can have no operators in common except similarity- transformations. 2. Now let A be any operator of G, and R of the group H, in K. Then since H is an invariant subgroup, A~ 1 RA = Ri belongs to H. Again, since G is an invariant subgroup, RAR~ 1 = Ai belongs to G. From these two equations we obtain RA=AR\ = A\R, and therefore 162 FINITE COLLINEATION GROUPS RiR- l = A~ l Ai. But R&- 1 belongs to Hand A~ l A l to G. It follows by 1 that RiR~ l = Ei, a similarity-transforma tion. Hence, R^E^R, so that A~ 1 RA=E 1 R. 3. Assuming R not to be a similarity-transformation, let it be written in canonical form, and let A represent in turn every operator of G. We can then prove by the process of 61 that G is not primitive. Accordingly, since this violates the hypothesis regarding G } the assump tion that H is contained in K is untenable. THEOREM 6. Let G be a self-conjugate subgroup of K of index h ( 28) and P a Sylow subgroup of G. Further more, let Q and Q be the largest subgroups of G and K respectively which contain P as an invariant subgroup. Then Q is a subgroup of Q of index h. Proof. Let g<i> and gh<l> be the orders of G and K; q and q the orders of Q and Q , and n the number of Sylow subgroups of the same order as P in G. These subgroups form a single conjugate set ( 36), and therefore q=(g^)/n. If A is any operator of K, then A~ 1 PA belongs to G and is of the same order as P ( 30, Exercise 2); the group A~ 1 PA is therefore the group P or another Sylow subgroup conjugate to P. Hence, the n subgroups conjugate to P in G also form a single conjugate set in K, and q = (gh<t>)/n, so that q f /q = h. Finally, Q is evidently a subgroup of Q by the definition of these groups. 119. Now, in the case of the Sylow subgroup (1), the group Q already has the maximum order 21</>, barring the existence of an invariant subgroup "H p ". The groups (D) and (E), 102, can therefore not be con tained as invariant subgroups in larger groups. The same argument applies to the group (F), when for P we take a group of order 81, which is already contained in a group Q of order 162 generated by P and V 2 . LINEAR GROUPS IN FOUR VARIABLES 163 There remain the groups (A), (B), and (C). Taking for P a subgroup of order 5, the group Q is here of order 10< (type (g), 111), and may be enlarged to a group Q of order 20< (type (h)). Hence, these groups (A)-(C) may possibly be enlarged to groups of twice their orders. In fact, the student may verify that the transformation* F : zi = K z 2 = K a* = ^J, * 4 = iK; *-^, v 2 does not belong to the group (B), but will transform the generators of (B) in the same manner as the substitution (ab) will transform the corresponding substitutions. Again, the transformation* F": does not belong to either (A) or (C), but will transform the generators there listed into new generators in the same way as the substitution (ab) will transform the corre sponding substitutions. We therefore obtain three new groups, respectively isomorphic with the symmetric groups on 5, 5, 6 letters, namely, (G) Group of order 120< generated by (A) and F". (H) Group of order 120< generated by (B) and F . (K) Group of order 720</> generated by (C) and F". Evidently none of these new groups can be enlarged. NON-PRIMITIVE GROUPS AND PRIMITIVE GROUPS WHICH CONTAIN INVARIANT NON-PRIMITIVE SUBGROUPS, 120-125 120. Intransitive and imprimitive groups. As remarked in 101, we shall not present an exhaustive analysis of the remaining groups in four variables. The problems involved are not very difficult, but need on occasion a mass of painstaking labor. * See Maschko, Mathematische Annalen, 1899, referred to in footnote to 102. 164 FINITE COLLINEATION GROUPS The intransitive groups in four variables fall into four classes, according to the number of variables in the different sets of intransitivity, namely, (1, 1, 1, 1), (1, 1, 2), (1, 3), or (2, 2) variables. To construct such a group, say one whose sets of intransitivity involve (2, 2) variables : (xi, x z ) and (#3, # 4 ), we select two transitive binary groups (chap, iii, (B)-(E)) that can be written as iso- morphic groups directly or after being enlarged by the method of 85. To the identity of one group, repeated say k times, will correspond an invariant subgroup of the other of order k (cf. 32). The transformations of these groups separately need not be of determinant unity (cf . 51, 2). Hence, the matrices of the groups (B)-(E) may first be modified by multiplying them by certain similarity-transformations (cf. 10, 12); moreover, operators in the canonical form (a, a, ft, /3), where a 2 /2 2 = 1, may even be added as new generators. Imprimitive groups are of two kinds: (a) groups which have the monomial form by a proper choice of variables, and (6) groups of the form discussed at the opening of 60. A group G of the first kind (a) is iso- morphic with one of the transitive substitution groups on four letters, namely, the alternating or symmetric groups on four letters, the Sylow subgroup of order 8 of the symmetric group, or the group (8), 43. Such a group is therefore of order 120, 240, 80, or 40, where represents the order of the invariant abelian subgroup which has the canonical form when G is written in mo nomial form. A group of the second kind (6) is generated by an invariant intransitive subgroup H and a trans formation T which permutes the two sets of intransi tivity of H. If the variables are so chosen that the matrices of H have the form C\, 14, that of T will have the form of the second matrix given in 60. LINEAR GROUPS IN FOUR VARIABLES 165 To evaluate the elements of T more definitely we may proceed as follows. The Sylow subgroups of H of order 3 a must be permuted among themselves by T, and since they already constitute a single conjugate set under H, it is possible to find an operator L in H such that the product LT transforms a given Sylow subgroup P into itself (cf. 107, 4). Writing P in canonical form, we find that the new T (viz., LT) is a monomial transformation, and we can even make such additional changes in the variables that T has the form Xi = x ^ x 2 = x\, and either x 3 = ax{, X4 = (3x!>, or x$ = a.x >, X4 = /3x[. Moreover, since any odd power of T permutes the two sets of intransitivity of H also, we may assume that originally such a power had been selected that the new T is of order 2 b . EXERCISE Prove that either (I), T has the form Xi = x 3 , x 2 =x i, x z = a-x{, x t = ax 2 , or else (II), to the group of similarity-transformations of one of the components of // embracing one set of intransitivity (xi, 2) will correspond an invariant subgroup of the other set (x 3 , x t ) of order 120, 240, or 600. (Hint: construct T 2 and T~ 1 PT, belonging to H, and make use of the facts that none of the groups (C)-(E), chap, iii, have a transformation of order 20 commutative with one of order 30; and also that if H contains a transformation whose multipliers are [0, 9, pw, pw 2 J and is of order in the variables (xi, z 2 ) and of order 30 in the variables (x 3 , x t ), then (II) is true.) 121. Primitive groups having invariant intransitive subgroups.* It is easily proved that if a primitive group G contains an invariant intransitive subgroup H, none of the sets of intransitivity can embrace just one variable (cf. method of proving the lemma, 61). The transformations of H must therefore have the form of * Goursat has determined all the groups in four variables which leave invariant the quadric xixt x-2X3=0. These include all the groups enumer ated in this and the following paragraph. Soo "Sur les substitutions orthogonales, etc.," Annales scientifiques de V&cole Normale Superieure, (3), VI (1889), pp. 02-79. 166 FINITE COLLINEATION GROUPS Ci, 14, and the two straight lines Xi = Xz = Q, x 3 = x 4 = are invariant under H. If these are the only invariant lines, G will be intransitive or imprimitive, since a line which is invariant under H is by an operator of G trans formed into a line also invariant under H. We therefore assume at least one additional invariant line; this can be written Xi+Xz = Q, x 2 +x 4 = by a suitable choice of variables. The matrices of H are then seen to have the form of the matrix (1), 104. All the lines invariant under H now belong to the family (2), 104, and are, as remarked, permuted by G. In order that this condition may be fulfilled, the matrices of G must necessarily have the form (7) pt pu qt qu pv pw qv qw rt ru st su rv rw sv sw The numbers p,q,..,wm the different transformations of G are readily determined from the fact that are corresponding matrices of two groups, say G\ and Gz, in two variables, simply isomorphic with G. (This we discover when we multiply together two transformations of the form (7)). Moreover, G\ and G 2 may simultane ously be given any convenient forms, independently of each other, that would be obtainable through separate changes of variables. They must both be primitive, and are therefore to be selected from the groups (C)-(E), chap. iii. For if, say, Gz were imprimitive, we could write it in monomial form; the group G would then appear as an intransitive or imprimitive group. LINEAR GROUPS IN FOUR VARIABLES 167 To the group of similarity-transformations in GI will correspond an invariant subgroup H 2 of (7 2 , and vice versa. The group H 2 is transitive, since it includes a component of H. Hence, if G% is of order 120, H z is of order 40 (generated by W\ and Wz, 57) or of order 120; if the order of G 2 is 240, that of # 2 is 40, 120, or 240; finally, if the order of G 2 is 600, that of # 2 is 600. Setting up an isomorphism with the primitive group G\, we find the orders of the respective groups as follows : joo 2 1 2 3 4 5 6 7 Hi: 40 40 120 120 120 120 240 240 600 Gi: 120 24<#> 12<#> 24< 12</> 12<A 24</> 24<#> 60</> H 2 : 4</> 4<#> 12<#> 12* 24c#> 600 240 600 600 G 2 : 120 240 120 240 240 600 240 600 600 G: 480 960 1440 2880 2880 7200 5760 14400 36000 The groups 1 and 2 are monomial, while the groups l-7 a are all primitive. No new types are obtained by interchanging the subscripts 1 and 2 in 3, 4, or 6, since the groups so derived are equivalent to the original groups. The primitive groups l-7 leave invariant the quadric surface 1X4 x 2 x 3 =Q, and this one only. The family (2), 104, constitutes one of the two systems of straight lines lying upon this surface; the other is 122. Groups containing as invariant subgroups the primitive groups of 121. An operator which leaves invariant each of the two families of lines lying on the quadric ZiZ 4 z 2 z 3 = will have the form (7); one which interchanges them will have the form pt qt pu qu pv qv pw qw rt st ru su rv sv rw sw 168 FINITE COLLINEATION GROUPS New groups, different from those already listed in 121, can therefore be obtained only by including a transformation T of the form (8) among the generators of the groups l-7. This operator will transform H\ into Hz and vice versa; the orders of Hi and H 2 are there fore equal, and we are limited to the cases 1, 2, 5, and 7. The order of T may be assumed a power of 2 (cf . argu ment at the end of 120). Under this assumption it is contained in a Sylow subgroup P of order 2 a+1 ( 39), namely, the group generated by T and the Sylow subgroup of G of order 2 a which is transformed into itself by T. Writing P in monomial form, we find that for T may be chosen a generator which permutes the variables in the order (xi)(x 2 x 3 )(x^), say T: xi = ax(, z 2 = #4 z 3 = y4, x = %x\ (aS = /3y). Now consider a group generated by T and the group 2. The groups HI and H 2 being given as in 57, (C), the conditions that T 2 belongs to G and that T~ l HiT = H 2 , are now found to be equivalent to the equations a 4 = /? 4 = y 4 = 8 4 . But since the change of variables indicated by the transformation (1, 1, i, i) produces the operators of G over again in the same forms (though not in the same order), we may introduce this change (if necessary) in T so as to obtain a 2 = /J 2 . Again, if a = ft, we may replace T by the product WT, where W belongs to P and has the canonical form (1, 1, 1, 1); if a= y, we replace T by TW. Hence, finally, we have the two possibilities: There result two groups, 8 and 9, both of order 576</>, generated respectively by 2 and T\ t 2 and T 2 . The transformation T$ is not contained in either 1 or 7. Hence we here obtain only one new group in each LINEAR GROUPS IN FOUR VARIABLES 169 case, 10 and 11, of orders 288< and 7200<, generated respectively by 1 and Ti, 7 and TV Finally, the group 5 already contains the transforma tion R= (1, 1, i, i) = TzT-i 1 . The groups generated by 5 and TI or by 5 and T 2 are accordingly identical, furnishing a single new group, 12, of order 1152<. EXERCISES 1. The primitive groups 1-12 all leave invariant the surface XiX* 2X3 = 0, and only this one of the second degree. Prove that any operator which transforms this surface into itself must have the form (7) or the form (8). Hence prove that a group which contains self-con jugately any one of the groups 1-12 is already included in this list. 2. The groups 1-12 and the group (B), 102, are the only primitive groups in four variables which leave invariant a quadric surface. In the case of the group (B), this surface has for equation Xi+x 1+2x3X4 = 0, which can be transformed into the equation ZiZi Ztz 3 = by the following change of variables: Zi = z 2 +z 3 , x 2 =i( 22+33), x 3 = Zi, Xt = 2zi. Introduce this change in (B) and compare the new generators with the matrix (7). 123. Primitive groups having invariant imprimitive subgroups. In the study of these groups the following proposition is found useful: Given a group G and a positive integer n. The group generated by the nth powers of the operators of G or of an invariant subgroup of G is again an invariant subgroup of G. (The method of proof is embodied in Exercise 3, 31.) It should be noted that the group generated by the nth powers of the operators of G is not necessarily the group generated by the nth powers of the generators of G. Now consider a group G which contains invariantly an imprimitive subgroup K of the kind classified under (6), 120. The group generated by the second powers of the operators of K is intransitive, and therefore G is found among the groups defined in 121. The only exception 170 FINITE COLLINEATION GROUPS is furnished by a group K the second powers of whose operators all reduce to similarity-transformations.. In such a case K must be of order 2 m < and can therefore be written in monomial form. We find by trial that it is of order 16< and is generated by the transformations (9) .(1,1, -1, -1), 0100 1000 0001 0010 A 2 =(l, -1, -1,1) 0010 0001 1000 0100 to which may be added the similarity-transformation (i, i, i, i). Next let K belong to the class (a), 120, at the outset. The cases where K is of order 4g or Sg come under the class of groups mentioned above ; there remain the cases where K permutes the variables in the same way as the alternat ing or the symmetric groups on four letters. The latter case is reduced to the former by taking for a new K the group generated by the second powers of the transforma tions of K as given. Again, when K corresponds to the alternating group, it is reduced to a monomial group of order 40 by taking the group generated by the third powers of its transformations. Hence, finally, our problem is limited to that of finding the primitive groups which contain the group (9) self- conjugately. 124. The group F of order 11520<t> isomorphic with the symmetric group on 6 letters.* Referring to 113, let G and G be equivalent groups in x\, . . and yi, . . , and H the isomorphic group in the six variables (6): * Klein, Mathematischc Annalen, II (1870), 198 ff.; IV (1871), 356; Maschke, ibid., XXX (1887), 496 ff. LINEAR GROUPS IN FOUR VARIABLES 171 Viz, , ^34. If we in the latter group change the vari ables to Wi, . . , w 6 , where (10) JJ 1 " 11+ ^" > the group (9) will have taken the canonical form and be of variety 6. Accordingly, H will permute among them selves the six variables wi, . . , w 6 (cf. 61). The following statements may now be proved by the student : (a) A transformation of G whose corresponding trans formation of H has the canonical form must belong to K. (b) Consequently, if two transformations of H per mute Wi, . . , w 6 in the same way, one of the two corre sponding transformations of G is equal to the other multiplied by a transformation of K. (c) Now, the transformations: -i i 0110 1001 - i /2 permute the variables Wi, . . , WQ in the orders and (wzWtWsWzWs) respectively, and will therefore with K generate a group F of order 16-720-<, isomorphic with the symmetric group on the six letters w^ . . , w & . By (a) and (6), this group will contain every transforma tion which leaves K invariant. (d) Hence, finally, all the primitive groups which contain K as an invariant subgroup are contained as subgroups in the group F. (e) Now, any two subgroups, F\ and Fz, are equiva lent when their corresponding substitution groups, F{ and F-2, on the letters w\, . . , WG are conjugate under the symmetric group of order 720. For, if a substitution 172 FINITE COLLINEATION GROUPS transforms F{ into F->, the corresponding operator of F will transform FI into F 2 . Hence, to determine all the primitive, non-equivalent subgroups of F, we first select a representative of each set of conjugate subgroups of the symmetric group under discussion. Every such repre sentative of order 5k will furnish a primitive group, since the linear group generated by K and T is primitive ( 101). There are in all 9 groups of order 5fc : Group Order Generating Substitutions 13 5-16* (ww^BMwO ; K, T; 14 10- 16* " (tiWeXwifc); K, T, R 2 ; 15 20-164> " (wwtwew,); K, T, R; 16 60-16< " (t*W4)(wi>); K, T, SB; 17 " " (uwOCuwe); K, T, BR; 18 120-164> " (w 6 w t ); K, T, A; 19 " " (wtwi) (w 3 w,) 20 360-16^ " (WiwMwiW*); K, T, AB; 21 720-16$ " (wiw 2 ) , K, T,S. Here A=p(l, i, i, 1), B = p(l, 1, 1, 1), where p (1 -\-i)/V%} and R is the transformation : x{= ~7^( 1 1 1 125. There are no more primitive groups. For a sub group F i of the symmetric group on six letters contains an invariant subgroup F 2 of order either 2, 4, 3, or 9, unless the order of F[ is a multiple of 5. If F z is of order 2 or 4, the corresponding subgroup F 2 of.F is of order 32< or 64<, and the group F\ containing F 2 invariantly belongs to the category of groups discussed in 121 (cf. 123). If F 2 is of order 3, it is generated by (w 3 w 6 W4) or -by (wiW 3 w<,) In the first case, FI contains a self-conjugate LINEAR GROUPS IN FOUR VARIABLES 173 intransitive subgroup, namely, one generated by the second powers of F 2 (the transformation corresponding to (wzWsWt) is BR AB and has the form of Ci, 14). In the second case, F 2 is generated by K and the trans formation Xi = x[, x z = 3, x 3 = x 4 , z 4 = xi, and FI by F 2 and the transformations B and SR 2 , corresponding to (wiW 2 ) (w 3 w 4 ) (W$WQ) and (^1^2) (W^WG) (wtfjuz), or is a subgroup of this group, excepting the possibility where the order of F\ is divisible by 9, treated below. But this group is monomial. Finally, if Fi is of order 9, the corresponding subgroup F 2 of F must be equivalent to the group 1, 121, both having an invariant subgroup of order 16<, equiva lent to K, and both being of the same order ( 124, (d)). Hence, F\ must be one of the groups 1-12. In conclusion, we point out that no new groups arise by enlarging any of the groups 13-21. For each of these groups contains a single invariant subgroup of order 16<, namely, K. This is readily seen when we observe that no subgroup of order 2 a of the symmetric group on the letters wi, . . , w 6 can be transformed into itself by T=(w 2 w^WQW S w 5 ). Accordingly, a group which contains one of the groups 13-21 self -con jugately must therefore also contain K self-conjugately, and is again one of the groups 13-21. CHAPTER VIII ON THE HISTORY AND APPLICATIONS OF LINEAR GROUPS* 126. The theory of linear groups of finite order may be said to have been originated by F. Klein, who in 1876 constructed the binary linear groups in order to solve a certain problem in the Theory of Invariants.! Sub sequently he extended the Galois Theory of Algebraic Equations by the introduction of linear groups t (cf. 127). An important problem connected with Linear Differ ential Equations, namely the determination of those equa tions of this type whose coefficients are rational functions of the independent variable x and whose solutions are algebraic functions of x, was in the meantime attacked by various men, in particular H. A. Schwarz, L. FuchsJ and C. Jordan. 1f Their solutions hinged upon the dis covery of the invariants of certain corresponding linear groups or the groups themselves, although the group- notion was not at first introduced, except by Jordan. This author made the problem entirely one of linear * In the matter of references, the following abbreviations are used in the present chapter: Annalen, 2, 9, 12, 28, 50, 52, 60, 71, for Mathematische Annalen, Bd. 2 (1870), 9 (1876), 12 (1877,) 28 (1887), 50 (1898), 52 (1899), 60 (1905), 71 (1912), respectively; Crelle, 75,81, 84, 85, for Journal fur die reine und angewandte Mathematik, Bd. 75 (1873), 81 (1876), 84 (1878), 85 (1878), respectively; Transactions, VI, VIII, XIV, for Transactions of the American Mathematical Society, VI (1905), VIII (1907), XIV (1913); Icosaeder, for Vorlesungen ilber das Icosaeder, by F. Klein, Leipzig, 1884; Valentiner, for De endelige Transformations-Cruppers Theori, Videnskabernes Selskabs Skrifter, (6), Copenhagen, 1889. t Annalen, 9, pp. 183-208. t Cf. Icosaeder. \\ Ibid., 81, pp. 97-142; 85, pp. 1-25. Crelle, 75, pp. 292-335. 1 Ibid., 84, pp. 89-215. 174 APPLICATIONS OF LINEAR GROUPS 175 groups; and, having proved a proposition concerning the order of such groups ( 74), he was able to state a general theorem about the degree of the algebraic equation whose roots satisfy a linear differential equation (cf. 128).* Having briefly mentioned the important applications of linear groups, outside of general group-theory proper, we shall give a survey of the .progress of the theory up to the present. As already stated, Klein began by the construction of the groups in two variables. This was followed by different determinations of the same groups by P. Gordan, Jordan, and H. Valentiner,f and of the groups in three variables by Jordan { and Valentiner. A general theory for any number. of variables was out lined by the former and applied by him with partial suc cess to the groups in four variables.il In addition, special groups of the latter category that arose in analysis, or special classes of such groups, were discussed by other mathematicians (Klein, ^f Goursat,** Bagnera,ff etc.). The complete determination of the groups in four variables (aside from intransitive and monomial types) was carried through by the author.Jt More recently, H. H. Mitchell has given a partial determination of these groups by means of a classification based upon certain geometrical prop erties^ There are, in the main, four distinct principles em ployed in the determination of the groups in 2, 3, or 4 * Ibid., p. 91. t Crelle, 84, pp. 125-215. t See footnote to 52. Valentiner (ct. footnote above). || Atti della Reale Accademia della Scienze fisiche e matematiche di Napoli, t. 8 (1879). H Annalen, 2, pp. 198 ff.; 28, pp. 504 ff., etc. ** Annales scientifiques del Ecole Normale Supcrieure (3), t. 6 (1889), pp. 9-102. ft Rendiconti del Circolo Matematico di Palermo, t. 15 (1901), 161- 309; t. 19 (1905), pp. 1-56. it Annalen, 60, pp. 204-31; Transactions, 6, pp. 232-36. Transactions, 14, pp. 123-42. 176 FINITE COLLINEATION GROUPS variables: (a) the original geometrical process of Klein (chap, iii); (6) the processes leading to a diophantine equation, which may be approached analytically (Jordan, 59), or geometrically (Valentiner, t Bagnera, Mitchell); (c) a process involving the relative geometrical properties of transformations which represent "homologies" and like forms (Valentiner, Bagnera, Mitchell; cf. 80, 103); (d) a process developed from the properties of the multi pliers of the transformations, which are roots of unity (Blichfeldt, 63-68). A new principle has been added recently by Bieberbach, though it had already been used by Valentiner in a certain form (see footnote p. 97).* Independent of these principles stands the theory of group characteristics, of which G. Frobenius is the discoverer (chap. vi). Important additions have been made by I. Schur, W. Burnside, and T. Molien.f Re cently L. E. Dickson has developed a theory of group characteristics for modular groups, t In conclusion we shall dwell for a moment on an important question connected with linear groups: What is the arithmetical nature of the elements involved in such groups? The following theorem has been established (Maschke, Burnside, II Schur 1f): "The n variables may be so chosen that every element in the matrices is a cyclotomic number (that is, a linear function of roots of unity with coefficients which are rational numbers). Possible exceptions to this rule can occur only if the * The author has amplified this principle and hopes shortly to publish his results (cf. 74). t See footnote to 84. t Transactions, 8, pp. 389-398; Bulletin of the American Mathematical Society (2), XIII (1907), 477-88. Annalen, 50, pp. 492-98. || Proceedings of the London Mathematical Society, (2), III (1905), 239. ^ Sitzungsberichte der Konigl.-Preuss. Akademie der Wissenschaften, 190G, pp. 164 fl. APPLICATIONS OF LINEAR GROUPS 177 characteristic equation (-0) n +A(-0) n - l + . . . = ( 23) of every transformation of the group has the form of a perfect kth power, k being greater than unity and a factor of n." In addition, Schur has proved that the ele ments are algebraic integers ( 134), when the variables are suitably chosen.* In the case of a monomial group, W. A. Manning has demonstrated that under a proper choice of variables, every non-zero element is a root of unity, f 127. We shall now outline briefly the main points in the Galois theory of equations and Klein s extension thereof.t Let A = x n -ciX n - 1 -\-C2X n - 2 - . . =0 be an equation whose coefficients ci, . . , c n are numbers in a domain R. (A domain R consists of all numbers which are rational functions with rational coefficients of certain specified numbers ki, . . , k m defining R; for instance, the domain defined by a single rational number k, different from zero, is the aggregate of all rational numbers, includ ing zero.) The equation A = is irreducible in R if A is irreducible; that is, if no factor of A exists which is a polynomial in x of degree <n and with coefficients in R. Under this condition, integers mi, . . , m n can be found such that the n\ expressions y\ t . . y n \, obtained from yi = niiXi+ . . +m n x n by subjecting x\, . . , x n to all the n\ possible permutations, are all distinct. Consider the function B=(yyi)(y y z ) . . . (y y n \). The co efficients of the different powers of y are all symmetric functions of xi, . . , x n and are therefore numbers in R. Let B =(y-yi)(y-y 2 ) . . . (y y g ) be a factor of B, irreducible in R; the equation B = is called a Galoisian resolvent of A = for the domain R. Its roots y\, . . , y a are obtained from one of them by subjecting x\, . . , x n * Annalen, 71, p. 365. t Bulletin of the American Mathematical Society (2), XII (1905), 77-79. I Of. Miller, Blichfeldt, and Dickson, Theory and Applications of Finite Groups, New York, 1916, pp. 279 fl. 178 FINITE COLLINEATION GROUPS to g substitutions forming a substitution group G, called the group of the equation A Q for the domain R. Then, according to the Galois theory, if a rational function of the roots xi, . . , x n with coefficients in R (or in an enlarged domain R containing R) is an absolute invariant under G, this function equals a number in R (or R ). Again, if G has an invariant subgroup of index p (a prime number), the resolvent B = Q can be broken up into factors by the " ad junction" to our domain R of the roots of unity of index p (i.e., including these roots among the defining numbers of the domain), as well as a radical l/K, where K is a number in the enlarged domain R. In Klein s theory, we construct the regular substitu tion group H of order g consisting of the permutations that take place among the roots of E = when x\, . . , x n are subjected to the substitutions of G. This group H is intransitive as a linear group ( 96), breaking up into component groups H , H", . . , of respectively n t n", . . variables, linear functions of y\, . . , y g . At least one of the numbers n , n" , , . is unity (say n f = 1), the corresponding variable being 2/1+2/2+ +2/0- If H contains an invariant subgroup of index p (a prime number), then an additional number n", . . is unity, say n"=l. The corresponding variable is of the form Y=y +0y"+ . . + Qp-iy (P-D, where is a root of unity of index p, and y , y", . . are each the sum of g/p of the letters y\, . . , y g . The transformations of H" are here of the form Y = O j Y , so that Y p K is an absolute invariant of H" and therefore also of H and G. In agree ment with the Galois theory, K is a number in the domain R obtained by adjoining to R, and we have Y = fy K. On the other hand, if H is simple, then none of the numbers n", . . equals unity. Consider in this case the group H" in n" variables. Its invariants are invariants of H and G, and are, accordingly, equal to numbers APPLICATIONS OF LINEAR GROUPS 179 in the domain R" obtained by adjoining to R those coefficients of the various products of x\, . . , x n in volved in the invariants in question which are not already numbers of R (these coefficients depend only on the in tegers mi, . . , m n and the " multiplication table" of the group H and are presumably "cyclotomic" numbers; cf. 95, 126). If therefore we are able to evaluate, in some way, the n" variables of H" from these invariants, we shall have obtained that many new unsymmetrical functions of the roots of A 0, and our problem has been reduced correspondingly. This evaluation is what is called the binary, ternary, etc., form-problem, in the cases n" = 2, 3, etc. The form-problem of the first order (n" = 1) is solved by the extraction of a radical (y K), and an equation can be "solved by radicals" if the form-problems of the successive resolvents are all of the first order. Consider the "general" quintic equation . . =0. If the numbers defining the domain R be the coefficients d, . . , c 5 and the square root of the discriminant of A = 0, the group of this equation becomes the alter nating group on 5 letters. The numbers ft", . . , corresponding to the different non-equivalent groups H", . . are 3, 3, 4, 5 (cf. Exercise 1, 97). The general quintic can accordingly be reduced by means of a ternary form-problem. But, if the equation A = is first thrown into the form A =v>-\-av 2 -{-bv-{-c = Q by means of the so-called Tschirnhausen transformation (requiring the extraction of a square root in addition to rational opera tions performed on the coefficients ci, . . , c 5 ), its reduc tion is made to depend upon a binary form-problem. For, the 60 functions obtained by permuting the roots v\, vz, . . , fs of A = 0, according to the alternating group on 5 letters, in the function 180 FINITE COLLINEATION GROUPS where 6 is a fifth root of unity, are all linear fractional functions of z\ (when account is taken of the relations namely the functions representing the 60 transformations of the linear fractional group (11) corresponding to the binary icosahedral group (E), 58 (second form). The invariants of this group are accordingly equal to numbers in the domain defined by 0, the coefficients of A = Q, and the square root of the discriminant of A = Q. Extending these results, Klein has made the reduction of the general equations of degrees 6 and 7 depend upon a ternary and quaternary form-problem respectively, the corresponding linear groups being the Valentiner group of order 360<* and a group of order 7!</2 in four variables first constructed by Klein, f Beyond that, the general equation of degree n^S can be reduced by a form-problem of order n 1, but not lower (Wimanf). 128. The application of linear groups to linear differ ential equations having algebraic solutions will now be briefly explained. By the " domain R" should here be understood the aggregate of all constants (real or complex, algebraic or transcendental) and all rational functions of a single complex variable x, and by an " algebraic func tion of x" shall be meant a function which is a root of an algebraic equation whose coefficients are functions in R. (11J Consider the differential equation (y = -r-, etc.): (1) y"+py +gy=0, where p and q are functions in R. Let it be given that the equation is " irreducible" in R (i.e., no solution of * Valentiner (cf. footnote at the opening of 126), p. 198; the group is (I), 82. t Annalen, 28, p. 519. J Ibid., 52, p. 243. APPLICATIONS OF LINEAR GROUPS 181 (1) satisfies an equation y +ry = Q, where r is in R), and also that two independent solutions y\, yz of (1) are alge braic functions of x. Then if s is an arbitrary constant (not specified), the function is a solution of the differential equation (2) v f +v*+pv+q = 0, and is at the same time a root of an algebraic equation (3) with coefficients Ai, . . in R. If we assume that (3) is irreducible in R, it follows that its roots are all solutions of (2). Hence, any such root must be of the form where s is a constant. On the other hand, the roots of (3) are all of the form v\ as regards s; it is therefore easy to prove that s=(as + 6)/(cs +d), where a, 6, c, d are certain (known) constants. But this is the typical form of a linear fractional transformation T 2 , and we may write (vi)Tz = vz. The roots of (3) are accordingly obtained from v\ by subjecting s to m linear fractional transformations Ti, . . , T m . It is readily seen that these transformations form a group G. For, writing s originally instead of s in vi, the roots of (3) would evidently again be derived from the new form of v\ (which is the original #2) by subjecting s to the same linear fractional transforma tions Ti, . . , T m . Accordingly, the products T 2 Ti, T\, . . y T 2 T m are Ti, . . , T m over again in some order. 182 FINITE COLLINEATION GROUPS It follows that m is one of the numbers g, 2g, 12, 24, 60 (cf. 11, 56-58). However, s can be specialized so as to reduce this number m. For, let G contain a subgroup H of order h, which possesses a linear invariant when it is written as a linear group in two " variables 7 s, t; by a suitable choice of these variables we can cause this invariant to be t. If then we put s = in (3), the resulting equation will have h roots all equal to 2/2/2/2, and (3) will break up into h equal factors. Omitting the case-m/A = 1, we finally find that the degree of the equation (3) may be put equal to 2, 4, 6, 12, in the cases where the "mono- dromie" groups are the linear fractional groups corre sponding to (B), (C), (D), (E), 56-58, respectively. Similar results are obtained for linear differential equations of any order. In the case of an irreducible linear homogeneous differential equation of the third order, whose coefficients are in R and whose solutions are algebraic, the degree of the algebraic equation with coeffi cients in R satisfied by the function y /y of a certain solution y of the differential equation, is 3, 6, 6, 9, 9, 6, 36, 21, according as the corresponding linear fractional group is isomorphic with the linear group (C), . . , or (J), 76, 79, 82. This is Jordan s theorem (cf. 126), as modified by adopting Painleve* s substitution (v = y /y).* The complete determination of the types of equations (1) whose monodromie group is the icosahedral group (E), 58, has been carried through by Klein, chiefly by aid of the "Schwarzian derivative" : y "/y - l(y"/y Y, which remains unaltered when y is subjected to a linear fractional transformation with constant coefficients, f Similar methods have been applied to the equations of the third order by Painleve** and Boulanger.J * Comptes rendua, Paris, 104 (1887), pp. 1829-32; ibid., 105 (1888), pp. 58-61. t Annalen, 12, pp. 167-80. I Journal de I E cole Polytechnique (2), 4 (1898), pp. 1-122. APPENDIX 129. Congruences. The symbol (1) A = B (mod A;) is read "A is congruent to B modulo fc" and takes the place of the equation A=B+Ck where C is a positive or negative integer or zero. The congruence is used in preference to the equation whenever it is immaterial just what the value of C is. The following rules apply: if A =B and C=D (mod k), then A-B=Q (mod/c), mA=mB (mod k) when m is an integer, AC=BD (mod/c), AC=BD (mod /c). The least absolute remainder (positive or negative) obtained when a number A is divided by k is called the remainder of A (mod k) . Remark. In chap, iv, the congruence notation (1) is extended to the case where A and B are sums of roots of unity, the meaning now being that "A equals A;X(the sum of a finite number of roots of unity)/ 130. Roots of a congruence. Let a, 6, c be given integers and p a prime number, then by "a root of the congruence " ax z +bx-\-c=Q (mod p) is meant such an integer (if any) which, when substituted for x, will cause the left-hand member to become an integral multiple (positive or negative) of p, or zero. 183 184 FINITE COLLINEATION GROUPS For example, the congruence x 2 1 = (mod 2) has for root every odd number. These are all =1 (mod 2); we therefore say that the congruence given has just one root (mod 2). If p>2, the congruence x 2 1 = (mod p) has two roots (namely 1). The congruence z 2 =2 (mod 3) has none. We have the following theorem: A congruence of the n ih degree (ax n + ... =0) has at most n roots (mod p), p being a prime number. 131. Indeterminate equations of the first degree. The congruence ax=c (mod b) is equivalent to the indeterminate equation ax+by = c, to be satisfied by integral values of x and y. If the highest common factor of a and b divides c, there is always a solution of this equation. In particu lar, if c=l, there is a solution if a and b are prime to each other. 132. On a certain class of determinants. The deter minant f(xi, . . . , x n ) = has the value f=(Xi-Xi) (Xt-Xi) . . - (Xn- (X n -Xz) ... (X 1 1 ... 1 Xi X2 ... x n xl xl ... xl z?- 1 xl~ l ... zr 1 APPENDIX and is a factor of the determinant 185 I /// . /// > // The value of the quotient F/f, when x\, . . . , x n all have been replaced by 1, may be found by treating the fraction F/f as an "indeterminate form 0/0." We put xi = I and let Xz approach 1 as a limit ; the value of F/f will then become Next we let x 3 approach 1, etc. The final result is the ratio of the two following determinants: 1 a a(a D o(o-l)(a-2) ... 1 b 6(6- D 6(6-l)(6-2) ... 1 m m(m 1) m(m l)(m-2) 1 1 1 . 1 2 2! 1 n-l (w-l)(w-2) . . . ( - 1)! We find namely D : [21 3! . D = 1 1 a b a 2 ... 1 m m 2 ... =/(a, 6, . . . , m). 186 FINITE COLLINEATION GROUPS 133. On roots of unity. A solution of the equation *-!, n being a positive integer, is called a root of unity. A solu tion a is in particular called a primitive n th root of unity, if n is the least integer for which a n = 1. In such a case n is called the index of the root. The following theorems are useful : 1. The product or ratio of two roots of unity is a root of unity. 2. Any positive or negative rational power of a root of unity is again a root of unity. 3. If n is the index of a root a, and m a positive integer, the index of a m is n/d, where d is the highest common factor of n and m. 4. If the index of a root is n = ab, where a and b are two integers which are prime to each other, then it is possible to find a root of index a, say a, and one of index 6, say ft, such that Q = afi. As is customary, we write <o, <o 2 for the roots of index 3, and i, i for the roots of index 4. 5. If a is a root of index n, then the n solutions of x n =l are 1, a, a 2 , . . , a n ~ 1 ) and we have 6. Theorem of Kronecker. * For the proper handling of a certain class of equations a very effective theorem due to KrOnecker is necessary. We shall not make a formal statement of the theorem, but explain its meaning by implication. Trie class of equations referred to are all of the form = 0, ai, . . . , a k being roots of unity, and the *"Memoires sur les facteurs irroductiblos de 1 exprossion x M -l," Journal de Afathtmatique pures et appliqutea, I, t. 19 (1854), pp. 178 ft. APPENDIX 187 question involved is this : If we know nothing about these roots except their number k, what can be inferred concerning their values? Kronecker s theorem implies that the k roots fall into sets, each containing a prime number of roots the sum of which equals zero. Moreover, if p is the number of roots in any one set, and if a is a root of index p, then the roots of the set are e, ea, ca 2 , . . . , eaP -1 , where e is an unknown root of unity. We shall discuss in full the cases k = 3, 4, 5. /c = 3: ai-f-a 2 -j- a 3 = 0. Here we have a 2 = a 1 co, a 3 =a 1 w 2 . k = 4: ai-j-a 2 +a 3 +a 4 = 0. We have two sets of two roots each, say i 4-^2 = 0, a 3 -f-a 4 = 0. ft = 5: ai-f-a 2 +a 3 -fa 4 -{-a 5 = 0. There are two possi bilities: one set only, or two sets containing 3 and 2 roots respectively. If J3 represents a primitive 5th root, and y, 8 roots of unknown indices, the two cases are respectively given by (y-y) + By means of Kronecker s theorem the following can be proved: 7. If N represents the sum of a finite number of roots of unity and k an integer, and if it be known that N/k is an algebraic integer ( 134), then N/k equals the sum of a finite number of roots of unity. More definitely, the roots in N can be arranged in two sets such that the sum of those in one set vanishes and those in the other set are each repeated k (or a multiple of k) times. 8. By Demoivre s Theorem: (cos 0+i sin 0) n = cos nO+i sin nO, 188 FINITE COLLINEATION GROUPS the roots of unity of index n can be expressed in the form a = cos 360+z sin- 360, n n where m represents in turn every integer prime to and less than n. The conjugate-imaginary of a m is accordingly a~ m . 134. On algebraic integers. An algebraic integer is a number which satisfies an equation of the form x n +aix n ~ l + . . +a n -ix+a n = 0, where a\, . . , a n are positive or negative integers or zero. If a and ft are algebraic integers, and k an ordinary integer, then a+/3, a/3, and ka are algebraic integers. If a/6 is an algebraic integer, and at the same time a and b are ordinary integers, then a/b must be an ordinary integer. INDEXES GENERAL INDEX (Numbers refer to pages] Abelian groups, 26; 43-45 Abstract groups, 30, n. Algebraic integer, 188 Alternating groups, 54; 60-61 Associative law, 5; 30 Binary groups, 63-75 Canonical form, 3; 24-27 Change of variables, 15-17 Characteristic equation, 27-28 Characteristics, 28; 117; of inverse and conjugate trans formations, and of substi tutions, 118; general theory, 116-38 Class of a substitution group is the least number of letters that are replaced by different letters by a substitution of the group (except the identity) Collineations and collineation groups, 10-12 Commutative law, 5; 31 Components of an intransitive linear group, 117 Composition of two groups, 125 Congruences, 183-84 Conjugate-imaginary groups, 18 Conjugate operators, sets, and subgroups, 36-38 Cycle of a substitution, 52 Cyclotomic number, 179 Degree of a substitution group is the number of distinct letters used in the substitutions of the group Determinant of a linear trans formation, 2; 13, exs. 3, 4 Differential equations having algebraic solutions, 180-82 Dihedral group, 70 Diophantine equation, 75 Domain, 177, 180 Equation of the fifth degree, 179-80 Equivalent groups, 64; 129; 135 Even substitutions, 53 Factor groups, 42-43 Finite groups, 33 Form problem, 179 Galois theory of equations, with Klein s extension, 177-80 Galoisian resolvent, 177 Generators, 9-10; 33; 39, ex. 3; 139, 2 Group characteristics, 116-38 Group-matrix, 133-35 Group of an equation, 178 Group of similarity-transforma tions, 13, ex. 1 Groups: of linear transforma tions, 7-15; of operators, 33; 117, 2; of substitutions, 54; 56-59; of order vft, 45-50; 80; 81; of order p*(f, 137; of the regular polyhedra, 69-73 ; leav ing invariant a quadric sur face, 169, exs. 1, 2 Hermitian form, 19 Hermitian invariant, 20-21 Hessian group, 109 191 192 FINITE COLLINEATION GROUPS Icosahedral group, 73 Identity, the, 3; 30; 51; 9; 33 Imprimitive linear groups, 76-79 Imprimitive substitution groups. 55 Index of a subgroup, 34 Intransitive linear groups, 17 Intransitive substitution groups, 55 Invariants, 120; 125 Invariant operators and sub groups, 39-40 Inverse: of a linear transforma tion, 5; 7, exs. 4, 5, 6; 9; 22; of an operator, 31; 32, ex. 5; 33; of a substitution, 51 Irreducible algebraic equations, 177 Irreducible differential equa tions, 180 Irreducible groups, 22-24 Isomorphism, 40-43; 117, 2 Linear fractional groups, 13 Linear groups, 8 Linear transformations, 1-7 Matrices of the transformations of a transitive linear group, 135, exs. 1, 2; 176; 177; sum and product of, 4; 119 Matrix of a linear transforma tion, 2 Monomial groups, 77; 80 Monodromie group, 182 Multiplication-table of a group, 40 Multipliers of a linear trans formation, 3; 7, ex. 8; 102, exs. 1, 2 Non-equivalent groups, 64; 135 Octahedral group, 72 Odd substitutions, 53 Operators, 1; 30 Order: of a linear group, 8; 82; 127; 129, ex. 2; of a linear transformation, 6; of an operator, 32; 35; of a group of operators, 33; of a sub group, 34; of a primitive linear group, 89; 92; 103 Permutations, 50 Power: of a linear transforma tion, 5; of an operator, 32 Primitive linear groups, 77; 94; 96; 101; 103 Primitive substitution groups, 55 Product: of linear transforma tions, 3-5; of matrices, 119; of operators, 30; of substi tutions, 51 Quaternary groups, 139-73 Quotient groups, 42-43 Reduced set, 23 Reducible groups, 22-24 Regular substitution group, 59; 131; 135 Roots of unity^ 186 Schwarzian derivative, 182 Self-conjugate operators and subgroups, 39-40 Set: of generators, 33; of non- equivalent component groups, 131 Sets: of conjugate operators, 36; of conjugate subgroups, 38; of imprimitivity (of a linear group), 77; of intransitivity (of a linear group), 18; of intransitivity (of a substitu tion group), 55 Similarity-transformations, 3; 7, ex. 2; 13, ex. 1; 18, ex. 1; 40, ex. 5 Simple groups, 39; 58; 60-61; 137; 138, ex.; 147 INDEXES 193 Subgroups, 8; 34-35; 46; 49, ex. 1 Substitutions, 50; written as linear transformations, 1 ; 118 Sum of matrices, 119 Sylow s theorem and Sylow subgroups, 46-50; 80; 147 Symmetric group, 54 Systems of imprimitivity (of a substitution group), 55. See also Sets Ternary groups, 104-15 Tetrahedral group, 71 Transform of an operator is the operator into which the given operator is transformed, 36 Transformations. See Linear Transformations Transitive linear groups, 17; 131 Transitive substitution groups, 55 Transposition, 53 Types of groups, 140 Unit circle, 94 Unitary form, 21-22; 24; 27 Valentiner group is the group (I), 113 Variety of a linear transforma tion and of an abelian group, 90 INDEX OF AUTHORS (Numbers refer to pages] Bagnera, G., 147, 175, 176 Klein, F., 4, 65, 142, 170, 174, Bieberbach, L., 97, 103,. 176 175, 176, 178, 180, 182 Blichfeldt, H. F., 29, 80, 102, Kronecker, L., 124, 186, 187 103, 115, 116, 147, 175, 176 Boulanger, A., 182 Lin S> G - H - 60 Burnside, W., 4, 29, 60, 80, 113, Loewy, A., 21 116, 123, 135, 137, 138, 176 . Manning, W. A., 177 Cole, F. N., 29, 60 Maschke, H., 112, 135, 141, 142, 159, 163, 170, 176 Demoivre, A., 187 Miller, G. A., 29, 60, 113 Dickson, L. E., 29, 30, 61, 116, Mitchell, H. H., 115, 175, 176 176, 177 Frobenius, G., 4, 97, 102, 103, 116, 119, 124, 176 Fuchs, L., 21, 65, 174 Galois, E., 177 Gordan, P., 65, 175 Goursat, E., 165, 175 Hermite, C., 19 Hilton, H., 29 Holder, O., 60 Huntington, E. V., 30 Molien, T., 1J6, 176 5. H., 21, 61, 141, Netto, E., 29 Painleve, P., 182 Picard, E., 21 Schur, I., 4, 103, 116, 119, 129, 176, 177 Schwarz, H., 174 Sylow, L., 46 Valentiner, H., 21, 65, 73, 97, 115, 175, 176, 180 Jordan, C., 4, 60, 64, 65, 73, 103, w 109, 115, 142, 174, 175, 176, Weber, H., 4 182 Wiman, A., 180 194 61 RETURN Astronomy/Mathematics/Statistics Library TO +> 1 00 Evans Hall 642-0370 LOAN PERIOD 1 1 MONTH 2 3 4 5 6 ALL BOOKS MAY BE RECALLED AFTER 7 DAYS UMMCT stmtef 3 " tMt * 15"UE AS STAMPED BELOW AUG 1 7 ZOO Recg UCB A/M/ JUL. 2 2001 OCT 2 4 2003 FORM NO. DD 19 UNIVERSITY OF CALIFORNIA, BERKELEY BERKELEY, CA 94720 C. BERKELEY LIBRARIES CDt 3 515771D TSSS?