Zhe •Qlnlversiti? of Cblcaflo FOUNDKD BY JOHN D. ROCKEFELLER THE TERNARY LINEAR TRANSFORMATION GROUP G3.360 AND ITS COMPLETE INVA- RIANT SYSTEM A DISSERTATIO SUBMITTED TO THE FACULTIES OF THE GRADUATE SCHOOLS OF ARTS, LITERATURE, AND SCIENCE, IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY (department of mathematics) BY GEORGE LINCOLN BROWN PRINTED BY Jlbc "GimvecsttB of dbicago press CHICAGO ITbe XflnlvergltB of Chicago FOUNDED BV JOHN D. ROCKEFELLBR THE TERNARY LINEAR TRANSFORMATION GROUP G3.360 AND ITS COMPLETE INVA- RIANT SYSTEM A DISSERTATION SUBMITTED TO THE FACULTIES OF THE GRADUATE SCHOOLS OF ARTS, LITERATURE, AND SCIENCE, IN CANDIDACY FOR THE DEGREE OF DOCTOR OF PHILOSOPHY (department of mathematics) BY GEORGE LINCOLN BROWN PRINTED BY {Tbc Tantvergitg ot Cbicago press CHICAGO EXCHANGE . AND ITS COMPLETE INVARIANT SYSTEM.' SECTION I. INTRODUCTION. The problem of verifying the existence of the coUineation group G360 in a definitive manner and of finding its complete invariant system was proposed to me by Professor Moore, of the University of Chicago, his suggestion being prompted by H. Valentiner's deterniinaticn^cf this group.* While the latter gives the method. by which the generators >if the group may be found, he does not compute the-r delin'ltivV forms/' ' After having deduced the complete invariant system under the direction of Professor Maschke, the solution of the problem was given by me before the Mathematical Club of the university, August 21, 1896. Soon after, my attention was called to an article by A. Wiman,' which had just been received at the university, and which is a discus- sion of this same group. I have thought it proper, therefore, after giving my own discussion, to indicate very briefly the interesting method which he employs to determine the complete invariant system, at the same time showing that it is equivalent to that found by myself. SECTION II. THE GENERATORS OF THE GROUP G3.360. Since the cube and the fifth roots of unity enter into the coefficients of the operators of our group, for convenience in calculation we make use of the fifteenth roots of unity, (« = o, I, 2, . . . 14) For these we have the reducing equation : derived from (^.5_i)(^_,) ^ 1 For literature on this subject see: H. Valentiner, De endelige Transformations- Cruppers Theori, pp. 192-8; A. Wiman, " Ueber eine einfache Gruppe von 360 ebenen Collineationen," Math. Annalen, Bd. XLVII, pp. S31-56; F. Gerbaldi, " Sul Gruppo semplice di 360 coUineazione pione," Palermo Rend., T. XII, pp. 23-94; T. XIII, pp. 161-99; R. Fricke, "Ueber eine einfache Gruppe von 360 Operationen," Gott. Nachr., 1896, pp. 199-206; L. Lachtin, "Die Differentialresolvente einer algebraischen Gleichung 6ten Grades mit einer Gruppe 36oster Ordnung," Math. Annalen., Bd. LI, pp. 463-72. 2 De endelige Transformations-Gruppers Theori, pp. 192-8. 3" Ueber eine einfache Gruppe von 360 ebenen Collineationen," Math. Annalen, Bd. XLVII, pp. 531-56 (issued August 16, 1896). 3 'sc^^sei 4 THE TERNARY LINEAR TRANSFORMATION GROUP 03.360 by means of which any one of them can be expressed as a linear func- tion of the eight roots, (« = O, I, 2, . . . 7) For the generators* of the collineation group G360 we have com- puted the definitive forms : fji.x' = X , U: fxy' = e^y , vx ^- ^—7^-^r + ' y + z . V: vy'= -=^x + ^ y J^ vz'= ^ ' X + ^ H 7^--2 , 2iri i/5 = 93(i_e3) (i_e6) . 6^ and Fare of periods 5 and 2, respectively, and for each of them the determinant of the coefficients is unity. Since we shall deal with the substitution group instead of the group of collineations, we choose ^3^ v3 = I, i. e.,\i.,v = i, 6^ or ^'°. If ii=6^ or 6^°, it is evident that i7 as a substitution has the period 15, while for the same values of v, V has the period 6 ; in either case the cyclic self-conjugate subgroup generated by x'=e^x, y' = e^y , z'=e^z, which groups we name G3, will belong to our group, and consequently each operator of the collineation group will appear at least three times in the substitution group, with the multipliers i, ^^ and 6^°. For the remaining case, /u.=:v = i, £/" and V as, substitutions have the same periods as when collineations. We shall find, however, that G3 appears 1 See Valentinhr, De endelige Transforntations-Gruppers Theort, p. 192. The notation of the operators has been changed from A and B to U and F respectively. 2 The notation G« will be used throughout this paper to indicate in general a group of order «, THE TERNARY LINEAR TRANSFORMATION GROUP 03.360 5 in the group in this case also ; ' and accordingly we shall take here- after, without further specification, ^^v = 1. We notice that the coefficients of V are all real, and that, if the complex conjugates of the coefficients of U are taken, U* is obtained. From this can be concluded : 1. The complex conjugates of the operators of our group also belong to the group. 2. The complex conjugates of the operators of any subgroup of our group also form a subgroup. 3. If any expression is invariant under the whole group, its com- plex conjugate is invariant, and consequently its real and imaginary parts are both invariant ; hence we may choose the invariant system real. SECTION III. THE ICOSAHEDRON SUBGROUPS OF G3.360. The order of G3.360 is determined by means of these subgroups. G3.360 is found to be isomorphic with the group of even permutations of six things. We combine t/'and Fby the general method for multiplying trans- formations and form e^X = -— y -\^ —^z , 1/5 1/5 VS ' 2 2 The second power of this operator is x'=^e^x , {U^VUVU^Vy.- y'=d^y , whence we see that our group contains G3 , and consequently the same operators as when /u. or 1/ is taken equal to 6^ or 6^°. If in the generators of the ternary icosahedron group as given by Professor Klein,^ I See sec. 3. a Klein, VorUsungen iiber das Ikosaeder, pp. 213-ig, 6 THE TERNARY LINEAR TRANSFORMATION GROUP 03.360 Ao == Ag , s.- a;=€A, , A^ =^ fr A^ , VIa:=a,+a^^a, , 27r» £ = * T = fl3 the substitution Ao=^x , A,= - iS^ky , A,=i6^^kz , be made, the operators £/" and {U^ VUVU^ Vy of our group are obtained. The group G^ is consequently a subgroup of our group. The invariant system of the G^o just mentioned is obtained by making the above substitution in the expressions' which are given by Professor Klein, and are : A^ = 3^ -{- kf" yz , B,= 8x*yz-2k'x'y'z'+k*y^z^-i-tk^x{y^- z^) , Q=^20X^y'z'— i6oA'x*y^z^-{- 2oJ^x'y*z*-{- 6k^y^z^-\- 4ik (y^ — 25) (32 x*- 20k' x'yz-\- s ^y' z') - k"" (7'° + 2") , D, = U(B^-y + ^""0) n 1(2 e^+ 2 ^') ^ + le^-ky - iO"- kz\ n\{2e^-\-2d'')x + ie^^ky -i6"^kz\ . V Corresponding to this subgroup is the complex conjugate subgroup which has as an invariant system the complex conjugates of these expressions, for which we shall use the notation, A^, B^, C„ Z>,. By successive applications of U and V, A^ is found to take in all eighteen expressions, six expressions which we shall call Ay (v = i, 2, , . , 6), and each of these multiplied by 6^ and 6^°. The manner in which these quadratic expressions are permuted by 6^ and Fcan be seen from the following table, in which U and V act upon the quadratic forms which are written in the upper horizontal line : I Ibid., pp. 217, 218. THE TERNARY LINEAR TRANSFORMATION GROUP €3.360 A, A, A, A, A, Ae u A, A, A, As A, A, V A. A, O^A, e-A, A, Ae the six quadratic expressions being A, = x'-\-k'yz , (v = 2, 3,. . .6) >> The complex conjugates of these quadratic expressions, A,, (v = 1, 2, . . . 6) , are permuted in exactly the same way by U'* and F. Now, every ternary transformation group for whose operators the determinants of the coefficients are unity, and which leaves a quadratic form unchanged, is holoedrically isomorphic with a binary collineation group,' and since there is no binary collineation group of order 6o« except for n — 1,'' the operators which leave any one of the quadratic expressions Ay unchanged must all be contained in the Ggo correspond- ing. Hence, by the general group theory, the order of the group generated by U and F is 60 X 18 = 1080. If U and Fare written as permutations of the six conies gotten by placing yi>' (v = i, 2, . . . 6) equal to zero, we have U={A^A,A,A,A,), F={A,A:){A,A,). These generate the group of even permutations of six things. The Geo to which Aj belongs is generated by C/={A,A^A,A,A,), {U^ VUVU^ Vf = {A^A,){A,A,) , which are generators of the group of even permutations of five things. It is evident, then, that the permutations of the five conies A^ {y=2, 3, ... 6) under the Ggo to which A^ belongs form the group of even permuta- 1 Weber, Lehrbuch der Algebra, Bd. II, Abschnitt vii, sec. 49, pp. 191, 192. 'Ibid., sec. 52, p. 203. 8 THE TERNARY LINEAR TRANSFORMATION GROUP 03.360 tions of five things with which group the Ggo is holoedrically iso- morphic. And since the six conjugate G^o are simple, they have no operator in common other than the identity, and, therefore, by the general theory of isomorphic correspondence,' the collineation group of U and Fis holoedrically isomorphic with the group of even permuta- tions of six things. We have found that, owing to the presence of G3 , every operator of the collineation group appears at least three times in the substitution group, with the multipliers i, 6^, and 6^°. Since the order of the sub- stitution group is 1080, each operator of the collineation group corre- sponds to only three operators in the substitution group. It is evident that this correspondence determines the meriedric isomorphism of the substitution group with the collineation group of U and V, to the identity of the latter group corresponding the group G3 of the former. To indicate this isomorphism we use the notation G3.360. The interesting, but not surprising, analytical relation existing between the two systems of quadratic forms belonging to the Ggo sub- groups can be seen from the identity {k'- 2) A, = A, -^6'°^ Ay . (f = 2. 3, ... 6) SECTION IV. THE INVARIANT SYSTEM OF G3.360 DETERMINED BY MEANS OF THE Gfio SUBGROUPS. From the nature of our group (since it contains G3') it is seen that every invariant under it must be of degree 3« (« = -f- integer). Also, since it contains Geo as a subgroup, its invariants must be integral functions of the expressions which form the invariant system of the Gfio subgroup. The invariants of G3.360 are consequently integral func- tions of A^, B^, Q, and Z>, (see sec. 3). Now, instead of B^ and C,, we can take as invariants of Ggo (V=I,2, ... 6) and C; = UAy ; (I'-a.a, ... 6) for from the way in which the quadratic forms are permuted under the group G3.360 it is evident that B^ and C are invariant under the Ggo, which leaves A^ unchanged ; moreover, it is easily determined that I See BOLZA, Om the Theory of Substitution Groups and its Application to Algebraic Equa- tions, sec. 8, art. 39. THE TERNARY LINEAR TRANSFORMATION GROUP 03.360 9 A^, B[ , and C/ are independent, since B^ is not the cube of A^, and C,' contains terms (for instance, in y^°, 2'°) which cannot be gotten by combining A^ and -5/ in an integral function. But B^ and A^ C/ are both invariant under G3.360 and independent of each other. We use for them, therefore, the notation T^g and i^',,, respectively. Every invariant of the sixth degree under G3.360 must have the form aAl-\-fiFe , where a and /3 are constants. A^ not being invariant under the main group, ^6 must be the only invariant of this degree. Likewise, every invariant of the twelfth degree can be written aAl-]-(3AlJ^,+ yFl + 8A,C\ . Now, since F^ and A^ C^ = F^^ are invariant, any invariant of the twelfth degree independent of these must have the form aA\ + pA\F, • but, since A\ is a divisor of this invariant form (if any such exist), and A^ under G3.360 is permuted with each of the other quadratic forms Ay {y = i, ■>,,.. . 6), the cubes of each of these other five quadratic forms must be divisors of the expression, which is impossible. Similarly, it can be shown that there are no invariants of the eight- eenth or twenty-fourth degree independent of Ff, and F[,. Again, it is evident that a new invariant independent of F^ and F[^ is given by >' = 1, 2, . . . 6 since this contains terms (for instance, in y° and z^) which cannot be gotten by combining F^ and F^^ . By applying the operator Voi our group to aAl-\-pB, = \F, and equating the coefficients of corresponding terms in the original and the transformed expressions, the ratio /8 15 -3^^^ is gotten, which yields, on removing the constant factor \= 8, Fi = x^-\- 15 x*yz— isx'y'z'- loy^z^— ^Vixiy^—z^) . The invariant F^^ has been given as the product of the six quad- ratic forms Ay (v = i, s, . . . 6). It can easily be verified by an inspection 10 THE TERNARY LINEAR TRANSFORMATION GROUP 03.360 of the terms that form the invariant of the twelfth degree, that the Hes- sian covariant of F^, which we shall call F^,, may be taken, and for the invariant of the thirtieth degree, the same function of F^,, which we name -^30 . We then have these invariants real. The F^^ is found to be F^^ == ar" — 1 8 x^^y z + 1 5 o^f 2° + 40 x^y^ 2^+345 x^f ^ + 156 oc'y'^ z^ — i%f z* — V I X (^s _ gS) [36 ;x:* + 1 2 x^y z + 1 20 oc'y'^— 3o>'3 23] — 3 (/° + 2") (^4-2^2) . On account of its length, we have not determined the expression fori^3o. We have proved the following relations : (i) ^Al=f^{F^=%F,, V (2) UA^=/,{F,, i^„)=^;. , V (3) ^Al=MF,, F^,), (4) ^Al=f,{F,, 7^„) , (5) ^A-=f,{F,, ir„) , V \ (6) ^A^^=f,{F,, i^„, f^) = f;, , where// (« = i, 2, ... 6) are integral functions of i^g, i^.^, and -^30. Con- sequently, every symmetric function of the cubes of the six quadratic forms Ay (v = i, 2, . . . 6) is an integral function of F(„ F„, and F^,. We can now prove that every invariant of even degree of G3.360 is an integral function of F^, F^^, and F^. For, let I^„ be any such invariant, then (l) /.n= 2 S 2 ^P-^^^^L^J p, c, T=o, I, 2, 3 . . . X^ X^ X^ [3 pp F'' A"-' 6p + 2^ is an integral function of J^6> -^12. and -^^30 of degree 2 (, being the sum of the products to the power t — o- of the six quad- ratic expressions in sets of five, and consequently an integral function of J^6, -^12 > and 7^30 of degree 10 (t — a). Substituting the values of the expressions (3) and (4) in (2) we get (5) ^-=3222 ^"'^ ^' ^" *^^^"'^ ^^' ' ^" ' ^-^ + p O- T and -F30, since i^^ is an integral function of J^e and ^,2 . Since D^ is the only fundamental invariant of odd degree under the Gfio to which it belongs, it must be a divisor of every invariant of odd degree under G3.360. The same is evidently true of ^^. There cannot be, consequently, more than one fundamental invariant of odd degree ; and if one such invariant exists, it consists of the fifteen linear factors of Z>j, together with the factors conjugate to these under G3.360. Now, since Z), remains invariant under a Geo and also under G3', or under a group G3.60, it cannot take more than six forms under G3.360. If we multiply together these six conjugate expressions, an expression of the ninetieth degree in x, y, z is obtained, which we call <^9o- In go each linear factor occurs as often as every other factor, and con- sequently ^go is the first, second, third, fifth, or sixth power of an invariant of G3.360 , according to the number of times each factor occurs. If each factor occurs an odd number of times, ^g,, is some power of an 12 THE TERNARY LINEAR TRANSFORMATION GROUP 03.360 invariant of even degree, and there is no invariant of odd degree. If each factor occurs six times, Z>, itself is invariant under G3.360, and also Z>i , which is impossible, since D^ is different from D^ . Consequently, if there is any fundamental invariant of odd degree, it must consist of forty-five linear factors, each of which occurs twice in <^g^. Now, we have an invariant of odd degree in the functional deter- minant of F^, i^j2, and F^„, which can easily be proved not identically equal to zero, and which we name dx '5 dx ^45- 9^« dy 3^. hy dy 8^6 8. Bz In Z>, and its conjugate Z>, we have twenty-five of the linear factors of i^45 , and, by applying the operators of our group, the remaining twenty factors can easily be found. It is evident that F^^, being of even degree, is an integral function of Fq, i^i2, and F^^. On account of the complexity of the expressions involved, we have not computed this analytical relation. We may, however, express F^^ in a very simple manner through the six quadratic forms which represent the Ggo conies of one system. For the points of intersection of these conies with each other lie on the lines of A.- A, gives us two of these lines. By applying the operators [/ and F to this difference, we can obtain the entire number of lines. This differ- ence takes forty-five forms under the group (neglecting the factors $^ and 0'°), viz., 'i < k; i,k = z,i, . , .6\ v = o, I, 2 / ('■ and, consequently, each linear factor is repeated, and we have F:, = TL{A,-6^^A,). V, i, k THE TERNARY LINEAR TRANSFORMATION GROUP 63.360 13 SECTION V. THE INVARIANT SYSTEM DETERMINED BY MR. WIMAN IS IDENTIFIED WITH THAT OF SECTION IV. CONCLUDING REMARKS. Since the operators of the coUineation group G360 have been related to those of the permutation group G360 by means of the permutations which C/'and F effect on the six Geo conies of one system, the subgroups of the coUineation group, and consequently those of the substitution group G3.360, can be determined through the subgroups of this permu- tation group. We have seen that F^, F^^, F^^, and F^^ can be expressed as integral symmetric functions of the six quadratic expressions which represent these conies. For the many interesting features of the sub- groups of the coUineation group G360 and their configurations, the reader is referred to the authors cited in sec. i of this paper. In concluding, it is merely our purpose to show briefly how Mr. Wiman determines the invariant system of the group G3.360, and that it is equiva- lent to that of sec. 4. The group G3.360 is reached by Mr. Wiman through the G^^ sub- groups. In the coUineation group G360 there are two systems each of fifteen conjugate G^^, and corresponding to these two systems each of six conjugate Ggo subgroups (see sec. 3), two Geo oi the same system having a Gi^ in common, which belongs to a G^ under the whole group. In the coUineation group G360 are forty-five operators of period 2, whose perspective axes form the two systems of self-conjugate triangles belonging to the G^^. These axes are evidently the lines given by (sec. 4) F,, = o . Each of the linear forms representing these axes is found to occur three times, with the cube roots of unity as factors. The presence of these roots indicates that the subgroup G3' (sec. 2) appears in the sub- stitution group. The normal form, of the invariant of the sixth degree is given by Mr. Wiman as Ce= 10:^373+ ^z{x^-\-y^)— 4S x'y^z'— is5xyz'-\- 2Tz\ and two Geo conies which belong to different systems are The normal form of the Ce above is transformed into F(, of sec. 4 by the substitutions 14 THE TERNARY LINEAR TRANSFORMATION GROUP €3.360 y =-y , X which also transforms the two conies corresponding into A^ and its conjugate ^^ (see sec. 3). After having made a study of the subgroup configurations of the collineation group G350 , Mr. Wiman affirms that on the curve F, = o are only three special point systems. These consist of the 72 points of tangency of the Geo conies of one system with those of the other system, and the 270 points in which F, = o is cut by the forty-five perspective axes, these points being divided into two closed systems of 90 and 180 points, respectively. Since the invariant of the sixth degree found by Mr. Wiman has been shown to be equivalent to that of see. 4, we may without con- fusion use for it here the notation Ff, . The Hessian H of F(, is given as the invariant of the twelfth degree, and the curve is found to cut out on F, = o the special system of 72 points. An invariant of the thirtieth degree independent of F^, and H is given as dy hH dyhx dzhx ^F, ^^F, h^F, ^ = dxdz dydz Szdy B'F, 80^ hH hH dif dx ^ dz the curve ^ = cutting out on F, THE TERNARY LINEAR TRANSFORMATION GROUP G^.^bo 1 5 the special system of 90 points doubly counted, the two curves having simple contact in each of these points. The functional determinant of Ff,, H, and $ is found to be an invariant of the forty-fifth degree ^, and, as has been stated, represents the forty-five perspective axes of the collineation group G^. By means of the point systems on F, = o it is now proved that every invariant of the group can be expressed through F^, H, ^, and *; also, that *' is an integral function of F^, H, and $. We have identified the Ff, given by Mr. Wiman with that of sec. 4 by a very simple transformation. From the subsequent steps in the two methods it is evident that the two invariant systems are equivalent. VITA. George Lincoln Brown was born January 25, 1869, in Bates county, Missouri. He entered the Preparatory Department of the University of the State of Missouri in September, 1885, and attended that institution irregularly during the next few years, receiving the degree of Bachelor of Science in June, 1892. He returned to the Missouri University as Teaching Fellow in Mathematics, and during the next two years pursued advanced work in this department under the direction of Professors W. B. Smith and W. C. Tindall, receiving the degree of Master of Science in 1893. During the years 1894-95 and 1895-96 he attended the University of Chicago as Fellow in Mathematics, and pursued graduate courses in this department with Professors E. H. Moore, O. Bolza, and H. Maschke, and with Dr. Kurt Laves in Mathematical Astronomy. Mr. Brown desires to express here his gratitude to the instructors mentioned above for their direction and words of encouragement, and to thank them especially for the inspiration which their lives have afforded him. UNIVERSITY OF CALIFORNIA LIBRARY BERKELEY Return to desk from which borrowed This book is DUE on the las. date stamped below. LD 21-100m-9,'48XB399sl6 Pamphlet Binder Gaylord Bros., Inc. Makers Stockton, Calif. PAT JAN. 21. 1908 6ir THE UNIVERSITY OF CALIFORNIA LIBRARY