GEOMETRY AND COLLINEATION GROUPS OF THE FINITE PROJECTIVE PLANE PG(2,2 2 ) A DISSERTATION Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy (department of mathematics) BY ULYSSES GRANT MITCHELL PRINCETON 1910 GEOMETRY AND COLLINEATION GROUPS OF THE FINITE PROJECTIVE PLANE PG(2,2 2 ) A DISSERTATION Presented to the Faculty of Princeton University in Candidacy for the Degree of Doctor of Philosophy ( d b p a r t m b nt of maths m a t i c s ) BY ULYSSES GRANT MITCHELL »RINCETON ■ 1910 WS Copies of this dissertation may be obtained on application to the University Library, Princeton, N. j. The price for each copy is 50 cents, which includes postage. ERRATA Page 6, line 21, add -j-a. y /i l:i - to the left member of the equation. Page ii, line 4, in place of "PG(2,2")" read 'TG(2,2 )." Page 12, line io, in place of 'Vs'— **-f-*8" read "px u '—ix t +x t " Page 12, line II, in place of "(/-J-p-f O" read " (p*-|_^-f-i) /• Page 1 8, line 30, in place of "0//" read "or." Page 20, line 23, In place of '7,* read U I 3 " (twice). - • ■ • * • • •• Press of The Journal- World Lawrence, Kansas 1914 2KK021 GEOMETRY AND COLLINEATION GROUPS OF THE FINITE PROJECTIVE PLANE PG (2,2-).* §1. Definition of a Finite Projective Plane. §2. Preliminary Theorems. §3. Types of Collineations in PG (2,2 n ). £4. Cyclic Groups in PG (2,2-). §o. The Group of Determinant Unit}' — Go 0160 - §0. The Group Leaving Invariant an Imaginary Triangle — G 63 . $7. Invariant Real Configurations and Their Groups. §8. Subgroups of the Group G 2880 Which Leaves a Line Invariant. §/. Definition of a Finite Projective Plane. • The definition and general properties of finite projective spaces together with references to the literature of the subject may be found in a paper by veblen and bussey in the Transactions of the American Mathematical Society, Vol. 7, pp. 241 -259. They used the symbol PG(k,p n ), where k,p,n are integers and p is a prime, to indicate a finite projective space of k dimensions having p n -|- 1 points to the line. It is the purpose of this paper to discuss some of the properties of the PG(2,2") and to determine all subgroups of the group of projective collineations in PG(2,2 2 ). We give a brief summary of the analytic and synthetic definitions of a finite projective plane. If x x .x.,,x. v are marks of a Galois fieldf [designated by GF(p n )] of order p n there are (p" n — l)/(p — l)=P" n -f-P n +l elements of the form (x t> x st jr,) provided that the elements (x^x.^x.^) and (Ix^lx^lx^) indicate the same element * Presented to the American Mathematical Society, April 29, 1911 + For definition and properties of a Galois field see E. H. MOORE, Subgroups of the Generalized Fjnite Modular Group, University of Chicago Dec. Pub, Vol. IX., pp. 141- 156; L. E. DICKSON, Linear Groups, pp. 1-14. 28802 t 2 U. G. Mitchell: Geometry and when / is any mark other than zero and provided that (0,0,0). be excluded from consideration. These elements constitute a finite projective plane if the equation M, X y -f - WoX 2 + « 3 *3= O [the domain for coefficients and variables being the GF(p n )] be taken as the equa- tion of a line except when u i =u 2 =u^=o. The line is denoted by the symbol (w,,h,,w 3 ) and the symbols (m„« 2 ,m 3 ) and (/« 1 ,/« 2 ,/» R ) where / is any mark other than zero denote the same line. The points of a line are those points whose co- ordinates (x ,,*,,*,,) satisfy its equation. Taking 0,1, i and r for the marks of the GF (2 2 ) where i is defined as a root of the equation r=;'+ / and hence r*=/ (mod. 2) the PG(2,2 : ) so defined may be exhibited in the table of alignment given on the opposite page. In the analytical processes of PG(2,2 n ) no distinction need be made between plus and minus signs since -1=1 (mod. 2). Synthetically a finite projective plane may be defined as a set of elements which for suggestiveness are called points, arranged in subsets called lines and subject to the following conditions: I. The set contains a finite number, greater than one, of lines, and each line contains p n -f-i points (p and n integers and p a prime). II. If A and B are distinct points there is one and only one line that contains A and B. III. All the points considered are in the same plane. From this definition it follows* that the principle of duality is valid in the plane so defined, that there are p n -\-l lines through each point and that the total number of points in the plane is p 2n +p n +l. In a PG (2,2 2 ) there are then 21 points and 21 lines. The following set of ele- ments arranged in 21 lines of 5 elements each will be seen to satisfy the given synthetic definition and to be identical with the table given opposite. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 . 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 G 1 2 3 14 15 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 11 12' 13 16 17 18 19 20 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 §2. Preliminary Theorems. Theorem 1. In PG(2,2 n ) the diagonal points of a complete quadrangle are collinear. Proof. Let three of the vertices, A, B and C of the quadrangle (Fig. 1) be taken as the triangle of reference and let the fourth vertex, D, be assigned the co- ordinates (1,1,1). The intersections of AB with CD, AC with BD and AD with * Cf. VEBLEN and YOUNG, Projective Geometry, Vol. I, pp. 16-17. © Ox 00 »■* 1 h "n! «^ ^ i oo © © © © © © © © CI © © © ►•1 CI © ©~ © © hi" CI ©" © © © © ©^ © © *1 ©~ CI »■■» © © ©^ •~i CI •> © ©_^ >N © © © © © ►x © © ©^ CI •"I © © CI © © © ©" •-T i*i © © © © © ©^ CI © © f © © : © CI •■^ i»i ©^ © © s © ~' ►x © d II ft X © II m X + X + • if © II ci X + •if © >>r *c ©" © II 03 + CI X ± .X~ ©- © 1 x" + CI •is © II H X + .x" © © + c» X + H © © II X © II « .X + X oo © © li + CI X + H ©\ © l X + CI .X + © © © II 03 X + CI + H © © II CI X + H © II n H + CI © © II M X + CI + It © II, © i CI H + H © n + © II 05 X CI CI X + x" oo © ll n X + CI X + .x" ©V © It H + x" 4 U. G. Mitchell: Geometry and 3C determine the diagonal points P,Q,R as (1,0,1), (i,J,o) and (o,/,/) respectively, which are collinear on the line x 1 -\-x 2 -\-x 3 =^0. B(oo0 F1G.1. Since the quadrangle A,B,C,D is projectively equivalent to any other quad- rangle in the plane the proof is complete. The line joining the diagonal points of any complete quadrangle will be re- ferred to as the diagonal line of the quadrangle. Definition of conic. A point conic is defined as the locus of the points of in- tersection of corresponding lines in two projective non-perspective pencils of lines. A line conic is defined as consisting of the lines that join corresponding points in two projective non -perspective ranges of points. In PG (2,2 n ) the number of points in a point conic is 2 n ~|-1, the number of lines in a pencil, and the number of lines in a line conic is 2 n -j-l, the number of points in a range. Hence in PG(2,2 2 ) a point conic consists of any five points no three of which are collinear and a line conic consists of any five lines no three of which are con- current. A tangent to a point conic is defined as a line which has one and only one point in common with the conic. There is one and only one tangent at a given point on a conic since there are 2 n -j-l lines through the point and 2 n lines join- ing it to other points of the conic. By taking the equations of two projective pencils of lines XP-\-/xQ=o and \P'-\-lxQ'=o [P=0, Q*=0, P'=o, Q'=o being equations in abbreviated no- tation of linej in PG(2,L ;: ) and A and /x marks of the GF(2 n )] and eliminating A and /x it is readily shown that the equation of a point conic is a homogen- eous equation of the second degree in three variables with coefficients in the GF(2 n ). Similarly, using line coordinates it may be shown that the equation Collin eation Groups of PG(2,2 2 ). 5 of a line conic is also a homogeneous equation of the second degree in three vari- ables with coefficients in the GB (2*). Theorem 2. Every equation of the form i>i where the coefficients are marks of the GF(.2 n ) is satisfied by the coordinates of at least one point in PG(2,2 n ). Proof. Suppose neither «,, nor a, 2 to be zero. Taking x 3 =i the equation reduces to a u x 1 2 -{-(a l2 x 2 -\-a i3 )x 1 -{-a 22 x 2 2 -\-a 23 x 2 -\-a 33 =o which is satisfied by and x 2 == — Moreover, since in the GF(2") every mark satisfies the a i2 n equation x 2 =* it is a perfect square and since — /=/ (mod. 2) its square root is unique. These values for x t and x 2 are therefore uniquely determined and lie in the GF(2"). If a lt =0, F(x l ,x._ i ,x. d )=o is satisfied by (i,o,o) and if (in ^t o and a lo =0 h (x^x^.x^^o is satisfied by ( \ ~, i 1 ' )- Theorem 3. Every equation of the form 1,2,3, F(x 1 ,x 2 ,xJ= S tfij*i*j=0 dgj,> i,j where x,,x 2 ,x 3 are point coordinates and the coefficients are marks of the GF(2 Ti ) represents a point conic in PG(2,2 n ). Proof. Since by the previous theorem F(x 1 x 2 ,x 3 )=o is satisfied by at least one point in PG(2,2 n ), by means of a linear transformation of that point into the point (o,o, i ) F(x^,x 2 ,x 3 )=o can be transformed into the equation F, (*,,*,,,*,) = b 11 x 1 2 -{-b 12 x l x 2 -±b 22 x 2 2 +b x3 x x x 3 +b 2i x ? x 3 =0 This equation may be written *i(*u*iH-*ia*a+*ia*s) + **(**2*iH-*i8*«)""0 which is seen to be the locus of points of intersection of corresponding lines in the two projective pencils of lines lxi+m (*u* l -|-*ia*a+*i3*a) =o and lx x -\-m (b 22 x 2 -{-b 23 x 3 )=o Hence F(x lt x 2 ,x 3 )=o represents the locus previously defined as a point conic. Theorem 4. In PG(2,2 n ) all tangents to the point conic 1,2,3, F(x lt x 2 ,x 3 )== 2 a, J x,jf ] =o (^i) i.j are concurrent and their point of concurrence is (a 23 /i l3 ,a ,.,). 6 U. G. Mitchell: Geometry and Proof. The line f^-f r 2 * 2 +f 3 * 3 ==0, the domain for variables and coefficients being the GF(2"), will be a tangent to the conic F ( Xl ,x 2 ,x s )=0 in case it has but one point in common with the conic. Eliminating x t from the two equations gives (a„ 2 +("u+* 22 )p+ A =0 C 1 ) which expresses the condition for consistency. According as (1) has one, two, or no roots in the GF(2 n ) T has one, two, or no fixed points in PG(2,2 n ) and is designated correspondingly as parabolic, hyper- bolic , or elliptic. In PG( 2,2") there are 2 n (2 n -l) equations of the form (1) since A is not zero and there are 2 n -l marks other than zero in the GF(2 n ). One half of these equations have both roots in the GF(2 n ) and the other half have no roots in the GF(2 n ). Of those having roots in the GF(2 n ), 2 n -l have coincident and (2 n -l) (2 n_1 -l) have distinct roots. The necessary and sufficient condition that (1) shall have coincident roots is that a n ^Etf 22 (mod.2). When tf n ==tf 2 2 is substi- tuted in T it is found that T 2 becomes the identical collineation and hence every parabolic transformation in PG(2.2 n ) is of period two. An hyperbolic transforma- tion permutes 2 n -l points and hence its period is 2 n -l or some factor of 2 n -l. Similarly the period of an elliptic transformation must be 2 n — (-1 or some factor of 2 n +l. Suppose (1) to be irreducible in the GF(2 n ). From the theory of Galois fields we know that- its roots then are marks p lt p* n of the GF(2 2n ) conjugate with respect to the GF(2 n ). Substituting these values in T we find the invariant points of T to be (« 12 ,a 1 ,-j-p 1 ) and (tz 12 ,a ll -\-p i 2t> ). There are, therefore, on the line 2 n (2 n -l) points of PG(2,2 2n ) (referred to as "imaginary"' points) arranged in 2 n_1 (2 n -l) pairs of conjugate points which figure as the double points of the elliptic transformations. Hence in PG(2,2 2 ) there are six such pairs on each line. Ar. in ordinary projective geometry it can be proved that any three points on the line in PG(2,2 n ) can be transformed into any other three points of the line and that if three points of the line are fixed all points of the lin** are fixed. From this it follows that the number of parabolic transformations is 2 2n - 1, of hyperbolic transformations is 2"- 1 (2 n -2) (2 n -f 1) and of elliptic transformations is 2 2n (2 n -l)/2. The total number of collineations on the line is the total number of distinct trans- formations T having coefficients in the GF(2 n ) and determinant not zero. This Is determined to be 2 n (2 2n -l). In PG(2,2 2 ) according to the above there ore 60 transformations on the line and of these 15 are parabolic, 20 hyperbolic and 24 elliptic. Since the total group is of order 60 and can be exhibited as permutations of the five points df the line, it must be the alternating group on five symbols and hence its subgroups ere well known. They will, however, be enumerated later in determining the groups which leave a line invariant. Transformations in the plane. The general linear homogenous transformation in PG(2,2 n ) may be written J, : P x i '=a il x l -\- a i2 x 2 +a i3 x 3 , i=i,2,3, 8 U. G. Mitchell: Geometry and where the determinant A=|«u| of the transformation is not zero and the domain for the coefficients is the GF(2 n ). To determine the invariant points we set x\=x u (i=l,2,3) and obtain the characteristic cubic p > + (« n -M„+«3s) p 2 + (^u+^m+^ss) p+A=o (2) where A n is the co-factor of a^ in A—|tfjj|. Since there are 2 n marks in the GF(2») there are 2 2n (2 n -l) equations of the form (2) belonging to the GF(2 n ). Of these 2"-l have all three roots coincident, (2 n -l) (2 n -2) two roots coincident and the other distinct, (2 n — 1) (2 n — 2(2 n — 3)/3 \ all three roots dis- tinct, and 2"(2 n -l)7'2 one root in the GF(2 n ) and two roots in the GF(2 2n ) but conjugate with respect to the GF(2 n ). These last are made up of the products of the 2 n — 1 linear factors in GF(2*) with the 2"(2"- 1 )/2 irreducible quadratics which appeared as characterictic equations of transformations on the line. There are then 2" (2»— 1)(2 B+1 — l)/3cubics (2) having roots in GF(2") or GF(2-»). The remaining 2 n (2-"-l)/3 are irreducible in the GF(2") but from the theory of Galois fields* it follows that their roots are marks of the GF(2 3n ) conjugate with respect to the GF(2 n ). If, therefore, A be a root of an irreducible cubic belonging to the GF(2 n ) its other roots are A 2 " and A 2 "", where A is a mark of the GF(2 3n ). If we put Xi'=x u ( ;=1,2,3) in T, and substitute A for p we obtain Xl :^,:at 3 =(^ ]1 +<7 2: ,A+^,,A+A'') :(4 12 +a 21 k) :(// 1 ,,+« ;1 A) as the corresponding invariant point. The other invariant points are then neces- sarily the points obtained by substituting for A in this expression A" and A :; re- spectively. Hence every transformation T, whose characteristic equation (2) is irreducible in the GF (2 n ) leaves invariant a triangle in PG(2,2 3n ) which will be designated as an imaginary triangle to indicate that it is not in PG(2,2 n ). Corresponding to the three cases in which the characteristic equation (2) has three distinct roots there are then three types of transformations having for in- variant figure a triangle. These will be designated as type I,„ type I,, type I 3 ac- cording as the invariant triangle has none, one, or three of its vertices in PG(2,2 n ). If the equation (2) has two roots coincident the corresponding collineation is designated type II and leaves invariant two points and by duality two lines. Its invariant figure has two points on one line and two lines on one point. If the three roots of (2) coincide the corresponding transformation leaves invariant a lineal element and is called type III. Two special cases of the«e will be classified as separate types because of their importance. A transformation other than the identity which leave* invariant all points of a line / called the axis and all lines through a point P called the center, is called a homology or type IV. Such trans- formations appear among the powers of those. of types Ij and II. A transforma- tion, other than identity, which leaves invariant all points of a line / and all lines through a point P on / is called an elation or type V. The point P and the line Cf. Dickson, 1. c, p 21 and p. 53. Collin eation Groups of PG(2,2 2 ). 9 / are called the center and axis, respectively, of the elation. Such transformations appear among the powers of those of types II and III. The invariant figures of the different types are shown in Fig. 2. Fixed lines which are imaginary are dotted and fixed points which are imaginary are left open. \/ / \ m The following formulae* for the number of transformations of each type in PG(2,2 n ) are readily obtained: Cf. Dickson, 1. c., pp. 237-239. 10 U. G. Mitchell: Geometry and Nl =2< n ( 2 2 "— i) 2 /3 • Nl 1 =2*"(2 3n — 1) (2 n — 1) 7 2 Ni 3 =2 3n (2 2n -f-2 n +l) (2"+l) (2 n — 2) (2 n — 3) /« Nn=2 2n (2 3n — 1) (2M-1) (2 n — 2) Nni=2 n (2 3n — 1) (2 2 "— 1) Niv=2 2n (2 2n -|-2 n -f-l) (2 n — 2) Nv=(2 3n — 1) (2 n -j-l) Identlty=l Total=2 3n ( 2 3n — 1 ) ( 2 2n — 1 ) According to these formulae the order of the total group of PG(2,2 : ) is 60480 distributed as follows: Ni =19200 Ni 1 =24192 Ni 3 = 2240 Nn=lC080 Nin=3780 Niv= 672 Nv= 315 Identity= 1 Total=60480 The group of all projective transformations in PG(2,2 2 ) will be designated aS VJ60480' §4. Cyclic Groups in PG(2,2 2 ). We wish to determine in detail the path-curves and periodicity of each of the types in PG(2,2 2 ). In so doing we shall at the fame time determine all of the cyclic subgroups. Type I Consider the collineation T : px 2 '=x,-\-x 2 -\-x 3 The determinant of T is A=r. Its characteristic equation p 3 -\-ip 2 -\-i 2 p-\- r=0 is irreducible in the GF(2 2 ) and has for roots ^ 47 ^ r ' 9 ,^ 62 where v is a primitive root of the GF(2 8 )* and hence v 2i =i, v 63 =l. The invariant points cf T are, therefore, A„=(l,w 9 ,v 7 ), B ^-:(1,^V 28 ) and C ( ^(l,v 1 V 48 ). T is of period 21 and permutes the points or PG(2,2 2 ) in the order of their numbering in the table of alignment. There is, accordingly, some power of T which will transform any given point of PG(2,2 2 ) into any other given point of PG(2,2 2 ). To show that every collineation of type I in PG(2,2 2 ) is conjugate to some * For Galois field tables, see an article by W. H. Bussey, in the Bull. Amer. Math. Soc, Vol. XI, p. 27. Collin eatjon Groups of PG(2,2 2 ). 11 power of T, it isonly necessary to show that any triangle A= ( A, ,A 2 ,A 3 ) , B= ( A, 4 ,A 2 4 ,A 8 4 ) , CblVVi V e ) where X^A, are any three marks in the GF(2«) linearly independent with respect to the GF(2 2 ), can be transformed into A ,B ,C , re- spectively, by a transformation in PG(2,2 n ). The condition that A,,A 2 ,A 3 be lin- early independent with respect to the GF(2 2 ) is necessary because it will be ob- served that if the coordinates of the point A satisfied the equation a 1 x l +a 2 x 2 -\- n^c^'O those of B and C would also satisfy the equation 2 4 ya x x x -\-a 2 x 2 -\-a z x z ) 2 as (a^^a^-^a^) 2 ■«(« 1 ;r 1 -f ,u ( fl 21 A, +tf 22 A 2 -|-fl 2 3A3 ) ) " which raised to the 2 2 and 2 4 powers, are seen to be the conditions that T trans- form B to B and C and C . In (4) we may assign a 3i , a 32 and 33 arbitrarily in the GF(2 2 ) provided not all are taken as zero. We then have fl 31 Aj-(-fl3 2 A2+ a 33^3 ==vk some mark other than zero of the GF(2 6 ). Since A,,A 2 ,A 3 are three marks of the GF(2 6 ) linearly independent with respect to the GF(2 2 ) it is pos- sible to choose a n ,i l2 and a lZ within the GF(2 2 ) such that «,,A 1 +fl 12 A 2 -|-«i3A 3 is any mark of the GF(2 <1 .)* Accordingly we take «,,,«, 2 and a 13 such that a nK~\~ a \ 2^2~f- tf i3 A 3 =*> k " 7 «'nd similarly «._,,, a 22 and a 2 . A such that tf n A 1 -{-AtaA s - , r a s«^a =r k ~ 61 . The desired transformation T is thereby determined within the GF(2 2 ). Moreover the determinant of T is not zero since tf 1I A 1 - r -c 12 A 2 +tf ]3 A3=i; k - 7 tf 21 A,+a 22 A, -4- fl 2S A3=» k -". «si A, +a. i2 \ 2 +a 3 . A \ 3 =v* form a set of simultaneous non-homogeneous equations in A P A 2 and A 3 . The 21 powers of T form a cyclic subgroup of G 0048t and since the triangle ABC can be chosen in 2 8 (2*-l) (2*-l)/3 different ways there are 960 such conju- gate cyclic subgroups in G 60480 . Since the determinant of T is r the determinant of T 2 is i*=i and the de- terminant of T,, 3 is 1. The powers of T , then, which are also powers of T 3 and no others are of determinant unity. The group G 21 (cyc. I„) consisting of the 21 powers of T contains accordingly a self-conjugate subgroup of order 7 consisting of the 7 powers of T„ 3 . G„„ 4S1) must contain 960 such cyclic subgroups. Again, T 7 is of period 3. Hence, G 2! (eye. I ) contains a cyclic subgroup of order 3 which must also be self-cou jugate since no others powers of T„ than pow- ers of T 7 are of period 3. G 60480 must contain 960 such conjugate subgroups. * Cf. Dickson, 1. c, p. 49 12 U. G. Mitchell: Geometry and T 7 must permute all points of PG(2,2 2 ) in triangles since if it permuted any three collinear points among themselves it would leave invariant the line joining them. It will be seen later (in discussing the simple group G 1C8 ) that a trans- formation of type I of period 7 permutes among themselves seven points so re- lated that for every four of the points which are no three collinear the other three are the diagonal points of their complete quadrangle. Type l v The collineation T, : P x 2 '=x 3 has the characteristic equation (p-f-1) (p 2 +p+l) = whose roots are 1, u and u* where a is a primitive root in the GF(2 4 ) and hence « 5 =j and « 15 =7. The invariant points of T, , are Ajss (7,0,0), B A = (o^w^Q^ee (o,/,w 4 ), and T x is therefore of type I x with A ± for center (or invariant real point) and x 1 =0 for axis ,or invariant real line). T x is of period 15 and Tj 5 and T^ are homolo- gies. Accordingly, the group G 15 (eye. IJ consisting of the 15 powers of T 1 contains a self -con jugate cyclic subgroup of order 3 containing two homologies and the identity. Since the determinant of T 1 is i, T 1 3 ,T 1 6 ,T 1 9 ,T 1 12 , T t 15 and no other powers have determinant unity and are of period 5 with the exception of TV^bsI. G 15 (cyc. I x ) therefore contains a self-conjugate cyclic subgroup of order 5 consisting of these transformations. Ti 3 , which is of period 5, permutes the lines through A x in cyclic order and hence a point P x not on the axis has 4 other conjugates P 2 ,P 3 ,P 4 ,P ii such that no two of the points P 1 ,Po,P 3 ,P 4 ,P 3 are collinear with A,. Moreover, no three of the points P 1 ,P 2 ,P 3 ,P 4 ,P 5 , can be collinear, for if they were the line containing them would be invariant under T x 8 . Hence, P 1 ,P 2 ,P 3 ,P 4 ,P r> form a point conic having A x for outside point. Evidently Tj 8 leaves invariant three such point conies having A x for outside point and by duality three line conies having x l ^=o for outside line. Every collineation T/ of type l 1 in PG(2,2 2 ) is conjugate to T x or some one of its powers since there is in PG(2,2 2 ) a transformation S transforming any point P' and line /' into (7,0,0) and x 1 =o respectively and a transformation S u leaving the point (7,0,0) fixed and changing any pair of conjugate imaginary points on x l =o into the pair (o,7,m 4 ). The collineation (SS 1 )T/(SS 1 ) _1 must then be some power of T x . In discussing the one-dimensional transformations it was shown that in PG(2,2 2 ) there are six pairs of conjugate imaginary points on each line. Hence there are in PG(2,2 2 ) 21-16-6 =2016 conjugate groups G^fcyc.I,) each con- taining a cyclic self -con jugate subgroup of order 5 consisting of the transformations of period 5, and a cyclic self-conjugate subgroup of order 3 consisting of the hom- ologies. There are 2016 of the cyclic subgroups of order 5 but only 21 -16=336 of the subgroups of order 3 since the same subgroup of order 3 appears with every G ]5 (cyc.I 1 ) which leaves invariant a given center and axis. COLLINEATION GROUPS OF PG(2,2 2 ). 13 Type I 3 . If T 3 be a transformation of type I, it leaves invariant a real trian- gle A,B,C T 3 is fully determined by its invariant triangle and the transforma- tion of a point P into a point P' provided the points A,B,C,P andP' are no three collinear. Hence, (Theorem 4, Cor. 4) there are two choices for P' for a given point P. Accordingly T 3 is of period three and permutes the 9 points not on the sides of its invariant triangle in three triangles. It should be noted that any one of these triangles together with the points A,B,C form a set of six points no three of which are collinear and therefore constitute in six different ways a point conic and its outside point. Also that any one of these triangles and two of the points A,B,C form a point conic left invariant by T 3 and hence that T, leaves invariant 9 different non-degenerate point conies. If A,B,C be taken as the tri- angle of reference the two transformations of type I 3 which leave it invariant are T, : px 2 =i"x 2 and T 3 2 : px 2 =ix 2 Since any triangle can be transformed into any other triangle by a collineation within the PG(2,2 2 ) it follows that every collineation of type I, is conjugate to T 3 or T 3 2 . Since 21 -20 -16/3 1— 1120 different triangles can be chosen in PG (2,2 2 ) there are 1120 conjugate cyclic groups G 3 (eye. I 3 ). Type II. A collineation T 2 of type II leaves invariant two real points A,B, and a real line / (distinct from AB) through one of the points, say A. Two lines fixed through A make the transformation of lines through A of period three. One line fixed through B makes the transformation of lines through B of period two. T 2 is therefore of period 6, but only T 2 and T 2 5 =T 2 ' are of type II. T 2 2 and T 2 4 are homologies with B for center ?nd / for axis and T 2 3 is an elation with A for center and AB for axis. If we select A as the point ( 0,0,0 B as the point ( i,o,o) and the line / as the line x i =o we find that any point P not on AB or / can be transformed into any other point P' not on AB, I. PA or BB Taking P and P' as (/,/,/) and (/',/, o) T 2 is determined as T 2 : nx 2 '=x 2 px/=x 2 -4-x :i On the line x x =o T 2 interchanges the points (o,/,o) and (o,i,i). It is easily seen that T„ or T.r 1 is conjugate to anv other transformation T./ of type II in PG(2,2 2 )." Since the invariant figure can be chosen in 21 •10-8=1080 different ways and for a given point P, on / there are three choices for P,' it follows that ^oo48o contains 5040 cyclic groups G 6 (cyc.II) each containing a self -con jugate cyclic subgroup of order two consisting of T, 3 (an elation) and the identity, and a self -con jugate cyclic subgroup of order three consisting of T 2 2 and IV (hom- ologies) and the identity. It is to be noted, however, that each subgroup of order two is common to 16 (since there are 4 choices for B on AB and 4 choices 14 U. G. Mitchell: Geometry and for / through A) different groups G„ (eye. II) and hence that there are but 315 such subgroups. Also that each subgroup of order three is common to 15 different groups G 6 (eye. II) (since there are 5 choices for A on / and 3 choices for the pairing of lines through B in each case) and hence there are but 336 different such sub- groups of order three. Type III. A transformation T of type III leaves invariant a line / and a point A on /. Since one line through A is fixed the transformation of lines through A is parabolic. T 2 is therefore an elation and T* must be the identical trans- formation. T is consequently of period 4 and permutes four points, no one on / and no three of which are collinear, in cyclic order. Since the transformation of four points no three of which are collinear into four such points fully de- termines a projective transformation it follows that a transformation T of type ill is fully determined by any four such points which T permutes in cyclic order. The collineation of type III determined by permuting the four points (7,0,0), ( I > I >o)>(i>0,i) f .(i,i,i), no three of which are collinear, in cyclic order as named is f»*i'— *i T: px 2 '=x 1 -\-x 2 P x./=x 2 -\-x z T leaves invariant the point (0,0,1) and the line x 1 =0. It is readily seen that in PG(2,2 2 ) every collineation of type III is conjugate to either T or T 3 . Four noncollinear points A,B,C,D can be chosen in 21 -20 -16 :9/4 !=2520 dif- ferent ways. Each cyclic order determines a transformation of type III not a power of any determined by any other cyclic order and each transformation of type III permutes in cyclic order the points of four different quadrangles. It follows therefore that there are 2520-3/4=1890 cyclic groups G 4 (eye. Ill) in ^60480 eacn containing a self -con jugate cyclic subgroup of order two. It is to be noted that a subgroup of order two is common to 6 different groups G 4 (cyc. Ill) and hence that there are but 315 such groups. Type IF. Homologies. The homologies in PG(2,2 2 ) have appeared as the 336 cyclic subgroups of the 2016 G 15 (eye. IJ and the 5040 G 6 (eye. II). It was shown that the 336 cyclic G 3 (eye. IV) were conjugate under the group G 60 48o- A homology, as has been seen, is of period three and its path-curves are the straight lines through the center. The homology , ; x l / =ix 1 T : fix 2 '=x 2 may be taken as a canonical form. Type V. Elations. The elations have appeared as 315 conjugate cyclic sub- groups of order two in both the 5040 groups G 6 (eye. II) and the 1890 groups G 4 (eye. III). Each elation is of period 2 and its path-curves are the straight lines through the center. The elation fJ x/=x, T: px 2 =x 2 px./=x } -{-x 3 Collin eation Groups of PG(2,2-). 15 which has for center the point {0,0,1) and for axis the line x l ==o may be taken as a canonical form. §5. The Group of Determinant Unity — G 20160 . Theorem 5. In PG (2,2") every group G of order N which contains col- lineations of determinant not unity contains exactly N/3 collineations of determi- nant unity. Proof. It is obvious that the collineations of determinant unity in G form a self-conjugate subgroup G n . Suppose n greater than N/3 . If T be any collinea- tion in G but not in G n the products of T and T J by the n collineations in G n are 2n distinct collineations and G would contain 3n>N distinct collinea- tions which is contrary to hypothesis. Suppose n to be less than N/3 . G must then contain m> N/3 collineations of determinant d where d is either i or r. If T be any collineation in G of determinant d 2 the products of the m collineations of determinant d by T are m>N/3 distinct collineations of determi riant unity in G, contrary to supposition. Since n is neither less nor greater than N/3 it follows that n= N/3 The Group of Determinant Unity. By Theorem 5 the group G 60480 has a self-con- jugate subgroup of determinant unity of order 60480/3=20160. In §4 it was shown that all collineations of types I a ,III, V and those of type I of period 7 and type I of period 5 were of determinant unity. Hence the Group 20160 of determinant unity contains the following collineations: The identical collineation 1 All collineations of type I 3 2240 Those collineations of type Ii which are of period 5, (1-3 of the total number) 8064 Those collineations of type I which are of period 7, (3-10 of the total number) 5760 All collineations of type III 3780 All collineations of type V 315 20160 The group will be designated as above by G 20180 . It has been proved that in any PG(k,p n ) the group of all collineations of determinant unity is the maximal simple subgroup of all collineations in the PG(k,p n ).* Theorem 6. Every collineation in G 2inM can be obtained as a product* of elations. Proof. Let T be any collineation of type I 3 determined by the equation * Cf. VEBLENand Bussey, 1. c, p. 253 and Dickson, 1. c, p. 87. 16 U. G. Mitchell: Geometry and T(A 1 A 2 A 3 A 4 )=A 1 A 2 A 3 A 5 where no three of the points A 1 ,A 2 ,A 3 ,A 4 ,A 5 are collinear. Two elations E 1 and E 2 are determined by the following equations: Ei(AiA 2 A 3 A 4 ) = AtA 3 A 2 A 4 E 2 (A 2 A 3 A 4 A 5 )=A A 2 A 5 A 4 such that their product E 2 E 1 =T. That E t and E 2 are elations follows from the facts that elations are the only collineations in PG(2,2 2 ) of period two and that the points A 1 ,A 2 ,A 3 ,A 4 ,A 3 are no three collinear. Since the five points are no three collinear they form a conic and since E 2 interchanges four of the points by pairs it leaves invariant point by point the diagonal line of the complete quadran- gle of the four points. By Corollary 5 of Theorem 4 this line contains the fifth point of the conic. E 2 , therefore, leaves Ai invariant and it is clear that E 2 E L -= T. Hence every collineation of type I 3 can be obtained as the product of two elations. Let Tj be any collineation of type I, of period 5 determined by the equation T x (A 1 A.,A 3 A 4 ) = A 2 A 3 A 4 A 5 , where A 1 ,A 2 ,A 3 ,A 4 ,A 5 are five points no three of which are collinear. Two elations E/ and E 2 ' are determined by the equations E 1 '(A 1 A,A 4 A,)=A 5 A 4 A 2 A 1) E/ ( A 2 A a A 4 A 5 ) = A 5 A 4 A 3 A 2 , ' such that E 2 , E/=T ] . That E/ is an elation leaving A 3 invariant and that E 2 is an elation leaving A 1 invariant follows by the reasoning given above to show that E 2 was an elation leaving A 1 invariant. Hence every collineation T l of type I x of period five can be obtained as the product of two elations. The transformation T : px/=Ari-|-*, is of type I of period 7. It is found that T = T X E where E is an elation, E: ox./=x., and T x is of type I 2 of period five px/-=x 1 -\-x., T, : ,'.v./=x 3 pX./=X.,-\-X :i But since every collineation of type I x of period five can be obtained as a product of elations and T or one of its powers is conjugate to every collineation of type I of period seven within the PG(2,2 2 ) it follows that every collineation of type I of period seven can be obtained as a product of elations. Let S be any collineation of type III determined by the equation S ( A x A 2 A 3 A 4 ) = A 2 A 3 A 4 Aj , where no three of the points A 1 ,A 2 ,A 3 ,A 4 are collinear. Any transformation so determined must be of type III because in PG(2,2 2 ) transformations of type III COLLINEATION GROUPS OF PG(2,2 2 ). 17 and no others are of period four. Then S=E 2 E 1 where E t and E, are elations determined by the equations E 1 (A 1 A 2 A;,A 4 ) = AjA 4 A 3 A 2> K 2 (A 1 A 2 A 3 A i ) = A 2 A 1 A 4 A 3 . E! and E 2 are again necessarily elations because elations are of period two and the points are no three collinear. Since every collineation not an elation in G 20 i 60 must be of type I 3 , Ij (of pe- riod 5), I (of period 7), or III the theorem is established. Theorem 7. In PG(2,2 2 ) if a group G a of determinant unity be trans- itive on all points and lines of the plane and contain a single elation it contains all elations. Proof. In PG(2,2 2 ) three and but three elations have the same center and axis since there are but three ways in which the four points other than the center on an invariant line can be paired. The theorem will follow, therefore, if it can be shown that if G a contain a single elation it must contain elations such that for any given line / and point P on / there are three elations in G a having P for center and / for axis. From the transitivity of G a it follows that G a must contain transforms of the given elation such that every point in the plane is the center and every line the axis of at least one elation. Also the order of G a must be a multiple of 21 and therefore G a must contain a collineation of period 3. Since the only collineations in PG(2,2') of determinant unity and period three are of type I 3 it follows that G a must contain collineations of type I 3 such that every point in the plane is a vertex and every line of the plane is a side of the invariant triangle of at least one collineation of type I 3 . For the given line /, then, there is in G a an elation E having / for axis and a collineation T of type I 3 having / for an invariant line. Four cases may arise, (a). P may be the center of F and an invariant point of T. Since T leaves invariant a point which E transforms they cannot be commuta- tive and hence TET -1 and T-ET 2 are the other two elations having P for cen- ter and / for axis, (b). P may be the center of E and not an invariant point of T. Let A and B be the invariant points on / of T. A must be the center of some elation Ej. If / be not the axis of E t we have E 1 TE 1 _1 =T 1 a collineation of tjpe I 3 having A and some point B' different from B on / for invariant points. By- transforming T, through the power of T which transforms B'to P the case is reduced to case (a). A similar argument applies to the point B. If neither A nor B be the center of an elation whose axis is not /, by case (a) G a must contain all elations having A or B for center and / for axis. The three elations E,, E 2 , E s having A for center and / for axis form with the identity a group since the product of any two of them is an elation having / for axis and A for center. Similarly the three elations E,',F./,E., having B for center and / for axis form 18 U. G. Mitchell: Geometry and a group. The nine products EjE/ are all distinct since if E i E j , =E k E 1 / it fol- lows that Ej'E/— EjEk which cannot be true. Moreover, every EiE/ is an elation having / foi axis since it is of determinant unity, leaves fixed every point of / and can not be the identity. Since the nine elations E s E/ are all distinct and have / for axis they include the three elations having P for center and / for axis, (c). P may not be the center of E and may be an invariant point of T. P must then be the center of some elation E' having some other line than / for axis. The transforms of E through E', T and T 2 give elations such that every point of / other than P is the center of an elation having / for axis. Since the lines through P can be interchanged by pairs in three ways only, the product of some two of these four elations is an elation with P for center and / for axis. This case is thereby reduced to case (a). (d). P may be neither the center of E nor an invariant point of T. If C, the center of E, be not an invariant point of T by transforming E through T or T 2 •'whichever transforms C to P) the case is reduced to case (b). If C, the center of E, be one of the invariants points of T we may transform E through E lt the elation having P for center and some other line than / for axis, and obtain an elation E 2 having some other point on / for center. If E 2 have for center the other invariant point of T the product EE 2 is an elation E 3 whose center is not one of the invariant points of T. The transforming of E 2 or E 3 through T or T 2 then reduces this case as above to case (b), and completes the proof of the theorem. Definition of Figure. In PG(2,2 n ) a point figure is defined as any set of m points where m is any positive integer less than 2 2n — |-2 n — (-1. Similarly a line figure consists of any m lines. The term figure is used to refer to either a point figure or a line figure. A real figure in PG(2,2 n ) is a figure all of whose points and lines belong to the PG(2,2"). It is obvious that any subgroup of G 00480 which leaves invariant no real figure is transitive on all points and lines of the plane. Theorem 8. There is no subgroup of G 20160 which does not leave invariant a real figure on an imaginary triangle. Proof. Any such subgroup G k can contain no elation, for by Theorem 7 if G k contained a single elation it would contain all elations and hence, by Theorem 6, all collineations in G 20]60 . Also G k can contain no collineation of type III since the square of a type III is an elation. Suppose G k to contain a collineation T, of type l x and let its center be designated P r As was seen in the proof of theorem 7, since G k is transitive and of determinant unity it contains some collineation T 3 of type I 3 which leaves P x invariant. Let l x and l 2 be the two lines through P x left invariant by T 3 . Since T 1 is of period 5 on the lines through P x some power of T 1( say T x m transforms / x to l 2 . Let / 3 ,/ 4 ,/ 5 be the lines into which T\ m tiansforms / 2 ,/ 3 ,/ 4 respectively. Some power of T 3 , say T 3 n , produces among the lines through Y > 1 the transformation (l x ) (I.,) (IJJ^)- Hence the product T, 2m T 3 n produces among the lines through P x the transformation (lj 3 ) (LI*) (l 5 ) COLLINEATION GROUPS OF PG(2,2 2 ). 19 The collineation T, 2m T 3 n leaves invariant the point P, and a single line /. through P r Such a collineation must be of type III or an elation. Hence G k can contain no collineation of type I r Since the only other collineations of determinant unity are of type I 3 (of period 3) and type I (of period 7) G k can contain only collineations of these two types. Since 20160— x2*'3 a '5r7 and the order of G k must be divisible by 21 the only possible orders for G k are 21 and 63. But as a consequence of Sylow's Theorem* any group of order 21 or G3 must contain a self -con jugate cyclic subgroup of order 7 since the order of the group can be written in the form 7m(l-f-7k) where 7m is the order of the largest group within which the cyclic subgroup of order 7 is self- conjugate and 1 -j-7k is the number of cyclic subgroups of order 7. For order 21 the only possibility is k=0 and m=3 and for order 63 k =0 and m=9. In PG(2,2-) the only possible cyclic group of order 7 is a G 7 which leaves invariant an imaginary triangle F,. But if the G- be self -con jugate within the G k every collineation in G k must leave invariant the imaginary triangle Fi. Hence there is no subgroup G k of G 201C0 which does not leave invariant either a real figure or an imaginary triangle. Theorem P. There is no subgroup of G 60480 except G 20]60 which does not leave invariant a real figure or an imaginary triangle. Proof. If any subgroup, say G n , exist it must contain a self-conjugate sub- group H n of determinant unity which leaves invariant no real figure or imaginary triangle contrary to theorem 8. §6. The Group Leaving Invariant an Imaginary Triangle — G 6S . Theorem 10. The group of all collineations in PG(2,2 2 ) which leave inva- riant a given imaginary triangle F t is of order 63. Proof. Let the group be designated G a . In §4 it was shown that if a collin- eation leave fixed one vertex of Fi it leaves fixed every vertex of F ( . Hence every collineation leaving Fi invariant must either permute the vertices of F| in cyclic order or leave each vertex fixed. It was also shown in § 4 that there are exactly 21 collineations — the 21 powers cf a type I () of period 21 — which leave each vortex of an imaginary triangle F t fixed. That there can not be more than 21 such col- lineations follows from the fact that ? collineation is fully determined by the leaving fixed of each vertex of an imaginary triangle and the transformation ot one real point into another real point. There can not be more than 21 collineations permuting the vertices AiBiCi of F| in a given cyclic order (AiBiC,). For suppose S,, S„ S„, S n to be n such collineations where n>21. Then if T be a collineation permuting Ai,B 1( Ci in the order (AiCiB t ) there are within G, n">21 collineations TSj,TS 2 , TS 3 ,TS n , distinct from each other and each leaving every vertex A|,B,,C| * See Burnside, Theory of Groups, p. 94. 20 U. G. Mitchell: Geometry and fixed, contrary to the hypothesis that G a contains but 21 collineations leaving each vertex of Fi fixed. That there exists a group of order 63 leaving F t invariant is shown by consid- eration of the transformations /»*i'=**i+*3 P x i'= x i J r x 3 T : P x./=x i -\-ix 2 and T 3 : ^v/=x 3 P x z '=x 2 -yr-x z p* 3 '=*2+*3 T is of type I of period 21 and T 3 is of type I 3 of period 3. T n leaves fixed each vertex Ai=(j,*; 27 ,v s6 ), Bi^(i,v iS ,v 18 ),Ci= (i,v r " l ,v 9 ) [where v is a prim- itive root of the GF(2 6 )] of Fi, and T 3 permutes these vertices in the order (A^Q). T and T 3 , therefore, generate a group of order 63 leaving invariant the imaginary triangle Fj. Since it was shown in §3 that F ( can be transformed into any other imaginary triangle Fi by a collineation within the PG(2,2'-), there is a group of order 63 leaving invariant any such triangle. Theorem 11. The only groups in PG(2,2 2 ) which leave invariant an im- aginary triangle Fi are the following: A. Groups leaving each vertex of Fi fixed. a. A cyclic group G 3 (cyc.I ) of collineations of type I of period J. b. A cyclic group G 7 (cyc. /„) of collineations of type I of period 7. c. A cyclic group G 21 (cyc. I ) of collineations of type I of period 21. B. Groups permuting the vertices of F%. a. A cyclic group G 3 (cyc. 7 ) of collineations of type I of period J. b. A cyclic group G 3 (cyc. I ) of collineations of type / of period J. c. An Abelian group G & leaving invariant also a real triangle, and contain- ing besides the identity 6 collineations of type I of period J and 2 col- lineations of type I 3 . d. A self-conjugate group G 21 of determinant unity containing besides the identity 6 collineations of type I of period J and 14 collineations of type I 3 . e. A group G 6Z of all collineations in PG(2,2 2 ) which leave Fi invariant containing besides the identity 6 collineations of type I of period 7, JO of type I of period J, 12 of type I of period 21, and, 14 of type I 3 . Proof. The existence of the group G„ a of all collineations in PG(2,2-) leaving l'j invariant was shown in the proof of the preceding Theorem. The existence of the cyclic subgroups is obvious and the existence of the G 21 of determinant unity follows from Theorem 5. The group G n is a Sylow subgroup and that it is Abe- lian follows from the fact that its order is the square of a prime.* To establish the Theorem it is only necessary to show further that every subgroup of the G 63 of all collineations leaving F s invariant is one of the kinds enumerated above. The only possible orders for such subgroups are 3, 7, 9, and 21. All subgroups of order 3 or 7 must be among the cyclic groups enumerated above since 3 and 7 are primes. A group of order 9 must be Abelian and by Theorem 5 must contain a G 3 (cyc. I 3 ) Cf. Burnside, 1. c, p. 63. Collin eation Groups of PG(2,2 2 ). 21 of determinant unity. But no such G 9 can contain more than one G 3 (cyc. I„) ; for if T, and T 2 be two collineations of type I, which do not belong to the same G 3 (cyc. I 3 ) the product of T t by the power of T 2 which permutes the vertices of Fi in inverse order is a collineation of determinant unity leaving each vertex of Fj fixed and therefore of type I of period 7. Hence every subgroup of G„ 3 of order 9 contains a self-conjugate G 8 (cyc. I 3 ) and leaves invariant a real triangle. A subgroup of G 03 of order 21 must be the direct product of a G 3 and a G 7 . Since the G T must be a G 7 (eye. I ) and the G., must be either a G 3 (cyc. I ) or a G, (eye. I 3 ) every such subgroup must be either a G 21 (cyc. I ) or a G 2l of determinant unity and therefore one of the kinds enumerated in the Theorem. § 7. Invariant Real Figures and Their Groups, It has now been shown that every subgroup of the G 00480 except the self-conju- gate G 2l)]00 of determinant unity leaves invariant a real figure or an imaginary triangle, and every group which leaves invariant an imaginary triangle has been determined. Accordingly we next take up the question of determining what real figures can be the invariant figures of groups in PG(2,2 2 ) and what group or groups leave each invariant. In determining these groups it is sufficient to con- sider point figures; for, since a collineation in the plane is self-dual, corresponding to every group which leaves invariant an //-line figure there is a group of the same order which leaves invariant the dual //-point figure. Abstractly considered the two groups are identical. Furthermore, in PG(2,2 2 ) it is sufficient to consider point figures in which the number of points n is less than 11, for if n == 11 the point figure consisting of 21— n (or some lesser number) can be taken as the inva- riant figure of the group. In this section will be determined all groups which leave invariant real point- figures whose points are not all collinear and which leave no point fixed under all transformations of the group. To obtain all such groups it is only necessary to determine for each value of n from n = 10 to n == 3 all groups which are trans- itive on all points of the //-point figure ; for, if such a group be not transitive on all points of an //-point invariant figure it must appear as a group which is trans- itive on an w-point figure where 3n. Also, since in PG(2,2 2 ) not more than 6 points can be chosen such that no three are collinear (Cor. 4, Theorem 4) if n > 6, k <£ 3. Since not more than 6 lines can be chosen such that no three are concurrent it follows that if m >6 either /<£3 or ra<£3£. By making use of the above relations and the fact that whenever m<« or 21 — m3 1 2 5 15 | 1 8 I 3 3 8 7 3 3 7 =F x G>2 6 2 3 9 1 2 2 1 6 1 ( 6 ! 2 4 12 — F ' 1 6>2 =F , 2 =F r „ 2 5 2 r V2 2 5 < 1 >P |2 l-F 1 3 ' 2 ' 3 6 1 5 1 2 4 10 K 1 2 2 4 It is observed that a group which leaves invariant an F^j also leaves invariant the figure made up of the remaining points and lines of the plane. This figu ; v will be called the residual figure and referred to as Rj„. We will consider these configurations in order. (a) 3 10 10 3 =-F L 10) Consider point P of F 10 , 3 and the three lines l x ,l^,l z of Ci. Veblen and Young, Projective Geometry, Vol. I, pp. 38-39. Collin eation Groups of PG(2,2-), 23 F„„ 3 which pass through P. On l x JJ :i are 7 points of F,,,,., and hence there are three points P lf P 8 ,P, of F, 0M not on any of the lines l^l.J^. This necessitates either that G lines l x ,iJ A , PP lt PP L „PP 3 pass through P or that two or more of the points P l ,P a ,P, are collinear with P. Since neither of these conclusions is allow- able under our hypotheses, F x0 „ is not a possible configuration in PG(2,2*). (b) 4 12 On each line of F , 3 must be two points which do not belong to F D , 3 . Let any line of F,„ be chosen as the line x l =o and the two points on it which do not belong to F 9 , 3 as the points 12 (o,o,j)* and 17 (o,i,o). Then the points (o,i,i), 10 (0,1,1), and 18 (o,i,i) on x,=o must be points of F , 3 . Through passes one and but one line which does not belong to F , 3 . Let the point of inter- section of this line with the similar line through 10 be chosen as the point 4 (1,0,0). Neither of these lines can contain any other points of F 9 , 3 than and -0, respectively, because the other three lines through either point contain six other points of F 9 , 3 which added to the three points on x x =o gives the total nine points of F n , 3 . The point 4 is therefore not a point in F 9 , 3 . Now no line through 12 which does not pass through 4 can be a line of F 9 , 3 since such a line contains at least three points (12 and its two intersections with the lines from 4 to and 10, respectively,) not in F 9 , 3 . A similar argument applies to the point 17. But every line of F 9 , 3 passes through some point of x x =o and but nine besides x t =o pass through the points 0, 10, 18. Hence the lines x 2 =o and x 2 =o are lines of F 9 „ and the nine points of F 9 , 3 lie three by three on the sides of the triangle of refer- ence On the side x 2 =o are the points 6 (i,o,i), 11 (1,0,1), 15 (i,o,i), and on x 3 =o are 3 (1,1,0), 7 (i,i,o), 19 (1, i,o). A reference to the table of alignment (p. 3) shows that these nine points are collinear by threes on nine other lines as shown in the accompanying figure (Fig. 3). Since x l =o Avas chosen as any line Fijj ure 3 * Numbers printed thus, 12, 17, etc., refer to the numbers assigned to points with certain coordinates, as given in the table of alignment on p. 3 24 U. G. Mitchell: Geometry and in F 0)3 it follows that through each point not belonging to F 9 , 3 pass two lines of F 11)3 and three lines not belonging to F s „ 3 , anu each une not belonging to F , a contains one and but one point of F 9 , 3 . Having found that F 9 , 3 is a possible configuration in PG(2,2-) we next pro- ceed to determine what collineations can leave it invariant. No line of F 9 , 3 can be the axis of an elation leaving F 9 , 3 invariant, for an elation interchanges all lines not invariant by pairs. Hence not more than one point of F , 3 can be invariant under an elation. But since an elation interchanges all points not invarianl by pairs at least one of the nine points of F 9)3 must be invariant under an elation which leaves F„, 3 invariant. If any point of F 9 , 3 is to be the center of such an elation the axis of the elation must be the one line through the point which contains no other point of F 9 , 3 , that is, the axis must be the line joining the point to the opposite vertex of the triangle of reference. An elation having such a center and axis and interchanging the other two vertices of the triangle of reference must leave F 9 , 3 invariant since the nine point* of F 9 , 3 lie three by three on the sides of the triangle of reference. It is obvious that there exists one and but one such elation* for each point in F 9 , 3 . Moreover, no point of the residual figure R 9 , 3 can be the center of an elation leaving F 9 , 3 invariant lor through such a point pass two lines of F 9 , 3 upon each of which are three points cf F 9 , 3 which could not be interchanged by pairs. There are, therefore, nine and but nine elations in PG(2,2 2 ) which leave F (l , 3 invariant. If T be a transformation of type I 3 which leaves F 9 , 3 invariant no point P of F,,, 3 can be a vertex of its invariant triangle; for at least one of the invariant lines through P would have to be a line of F 9)J and on that line a point of F 9 , 3 would be transformed into a point in R 9 , 3 . If any point in R , 3 can be a vertex of the invariant triangle of T the two lines through it belonging to F 9 , 3 must be the two invariant lines through the point, since otherwise at least one of them would be transformed into a line not in R 9 , 3 . The other two vertices of the triangle must be the two other points on these lines which do no*- belong to F 9 , 3 . Since the line joining these two points is a line of F 9 , 3 the points of F 9 , 3 are three by three on the sides of the triangle and hence T and T 2 leave F 9 , 3 invariant. Since but four such invariant triangles can be selected from the twelve points of R 9 , 3 there are eight and but eight collineations of type L which leave F 9 , 3 invariant. No transformation of type l 1 of period five or fifteen can leave F 9 , 3 invariant on account of its period. A collineation of type I 3 of period three is an homology. If an homology H leave F 9 , 3 invariant it can not have a point of F 9 , 3 for center since on one of the invariant lines a point of F 9 , 3 would be transformed into a point of R 9 , 3 . If any point P of R 9 , 3 can be the center of H, through P pass two and but two lines of F 9 , 3 and the axis of H must be the line / joining the two points of R 9 , 3 which lie on these two lines. Since / contains the other three points For example, if be chosen as the pomt, the elation must be px 2 '=r# 3 px/—ix a COLLINEATION GROUPS OF PG(2,2*). * 25 of F 0>3 a homology having P for center and / for axis must leave F 9 , 3 invariant. For each of the twelve points of R , 3 there are, then, two and but two homologies leaving F , 3 invariant. Accordingly, there are twenty-four homologies leaving F H , 3 invariant. If the homology H having P for center and / for axis be multiplied by an elation E leaving F 9 , 3 invariant and having some point Q on / for center a col- lineation T 2 is obtained which transforms F , 3 into itself and leaves invariant the points P and Q and the line /. Since T 2 is of determinant not unity and is of period 2 on / and period 3 on the line PQ it must be a collineation of type II. Since there are three and but three choices for the point Q on / there are six and but six collineations of type II having P for center which leave F 9 , 3 invariant. But every collineation of type II is of period six and has for its square a homology and for its cube an elation. Hence every collineation of type II which leaves F 9 „ invariant must be the product of a homology and an elation each leaving F 9 , 3 invariant and therefore related as were PI and E in obtaining T, above. Since the only points which can be the centers of homologies leaving F 9 , 3 invariant are the twelve points of R 9 , 3 and each homology and its square can be combined with three different elations there are seventy-two and but seventy-two collineations of type II leaving F 9 , 3 invariant. If F 9 , 3 can be left invariant by a collineation T 3 of type III, T 3 must have the same center and axis as some elation since T 3 2 is an elation. Taking for center and 4 for axis we find that there are six and but six collineations T 3 ,T 3 ', T 3 ", and their cubes, of type III which leave F 9 , 3 invariant and have this center and axis. This corresponds to the fact that there are only three ways in which the four lines of F , 3 through can be interchanged by pairs. The transformations T 3 , T/, and T 3 " are px 1 '=x 1 -\-Px 2 -\-ix 3 px/=x 1 -\-ix 2 -{-x :i p*i'='*i-f- ,J(, :.'-|- A: 3 T 3 : p*/=iAr 1 +/Ar 2 -|-/jf 3 17/ : p x 2 , =i 2 x 1 -{-ix 2 -\-ix 3 T 3 ": px./=ix l -\-x.,+ix 3 px/=i 2 x 1 -\-x 2 -{-ix 3 px/=x 1 -\-x 2 -\-ix 3 px 3 '=rx 1 -\-x 2 -\-x 3 That these collineations leave F 9 , 3 invariant is more readily seen when they are written in the form (points of F„, 3 in italics) : T,— (o) (ill IS 7) {6 iS 19 io) (1 16) (4 14) (5 12 9 17) (2 13 8 20) T 3 '={o) (J 19 15 6) (7 io ii 18) (1 14) (4 16) (5 13 9 20) (2 17 8 12) T,"— (o) (3 18 15 io) (6 7 19 u) (1 4) (14 16) (5 2 9 8) (12 20 17 13) Since these collineations are not commutative with the collineations of type I 3 and the elations which change the point into points on the other sides of the triangle of reference it is clear that the total group of collineations leaving F , s invariant must contain six collineations of type I 3 for each point of F yI , or alto- gether 54 such collineations. In fact, it is obvious that if any other center and axis than and 4 respectively had been selected six and but six collineations of type III leaving F 9 , 3 invariant and having that center and axis could have been deter- mined, provided that the center and axis selected were the center and axis of some elation leaving F 9 , 3 invariant. T x 05 T A X r# 2 i 2 x 3 X 3 *i X 26 U. G. Mitchell: Geometry and No transformation of type I of period 21 or 7 can leave F 9 , 3 invariant on ac- count of its period. If a transformation T be of type I of period three it per- mutes all points of the plane by triangles. As we have seen, there are four trian- gles each of which has for vertices points of R 9 , 3 and all other points on its sides points belonging to F , 3 . If T leave F 9 , 3 invariant it must transform this set of triangles into themselves, either by permuting the vertices of one of the triangles among themselves or by transforming one triangle wholly into another. Selecting one of these triangles, say 4 12 17, we determine all the transformations of type 1 of period three which permute its vertices in the order (4 12 17) and find that there are the six following: T T T T -"■01 ± 02 ■'■OS x 0- p x i= x 2 x 2 ix 2 x 2 fi X 2 == X Z 1X 3 X 3 X 3 px/=ix 1 x 1 x ± ix 1 Writing these in the form (points of F , 3 in italics) T 01 =(412 17) (019 u) (3 IS io) (6187) (12 9) (5 13 8) (14 20 16) T 02 =(4 12 17) (o 7 75) (3 11 18) (6 10 19) (1 16 5) (2 14 13) (8 9 20) T 03 =(4 12 17) (0 3 6) (7 11 10) (is 18 19) (1 8 14) (2 5 20) (9 13 16) T 04 =(4 12 17) (0 19 IS) (3 6 10) (11 18 7) (114 2) (8 13 9) (5 16 20) T 05 =(4 12 17) (o 3 a) (6 18 19) (7 15 io)(l 9 16) (8 20 14) (5 2 13) T 06 =(4 12 17) (07 6) (3 15 18) (11 10 19) (15 8) (2 20 9) (1314 16) it is seen that the vertices of no one of the triangles 2 8 16, 5 9 14, 1 20 14 are permuted by any of these transformations but that one triangle is transformed wholly into another. Since the squares of these transformations also leave F 9 , 3 invariant there are twelve transformations of type I of period three leaving F 9 , 3 invariant which permute the vertices of a given one of the four triangles whose vertices are in R 9 , 3 . There are, therefore, altogether 48 collineations of type I of period three which leave F 9 , 3 invariant. Summarizing, we have in the total group leaving F 9 , 3 invariant 9 elations, 8 type I 3 's, 54 type Ill's, 24 homologies, 72 type II's, 48 type I 's of period 3, and the identity, making a total of 216 collineations. The group is readily identified as the Hessian group G 21C discovered by C. Jordan in J 878.* Here the 9 points of F 9 , 3 represent the 9 points of inflection of each cubic of the pencil M*l 3 + *2 3 + *8 8 )+/**l*2*S=0 which is invariant under every collineation of the group. To verify that every point of F 9 , 3 lies on every cubic of the pencil it is only necessary to notice that every point of F 9 , 3 has one and but one coordinate zero and z 3 =(r) 3 =l. Every subgroup of G 210 leaves F 9 , 3 invariant and every group in * Crelle, Vol. 84, (1878) pp. 89-215. The group is defined by leaving invariant a pencil of cubics XF-4-//.H^O where F is a ternary cubic and H its Hessian covariant. Collineation Groups of PG(2,2 2 ). 27 PG(2,2 2 ) (except the G 21(1 itself) which leaves F,„ invariant must be a sub- group of the G216.* (e) Q 3 1' ■. =F , S '. Consider a point P of F„ s ' and the three lines /,./..,/., of -J— --___' F 0)3 ' which pass through P. On /,,/.,,/., are 7 points of F„, .' and there are, there- fore, two points P' and P" of F , z ' not on /,,/.,, or /.,. There are two possibilities only — P' and P" are or are not collinear with P. If P' and P" are collinear with P the configuration is F 9 , a except that ?■ lines are omitted. If P' and P" are not. collinear with P there must be two lines through P (namely PP' and PP") which contain but two points of F n ,.,' each. Since P can be transformed into any other point of F<„./ this would necessitate the existence of a configuration of 9 points arranged two points to the line which contradicts the condition that when the number of points exceeds G there must be at least 3 to the line. Hence there is but one possible arrangement and that is as the 9 points and 9 of the lines of F 8 , 3 . Since F 9 , 3 includes all lines joining its points it follows that every trans- formation which permutes the 9 points and 9 of the lines of F 9 , 3 also permutes «imong themselves the other 3 lines of F 9 , a . F,„ 3 ' is, therefore, the subgroup G M of order 54 of F 9 , 3 which leaves invariant a simple 3-line composed of 3 lines of F ,, jo chosen that no two of them meet in a point of F 9 , 3 . j 3 8 I =I r s>3' Let any line of F 8 , 3 be chosen as the line x 3 =o. Since but 6 lines of F 8 , 3 meet x 3 =0 in points of F 8 , 3 there is one and jut one line of F 8 , 3 which meets .v 3 =o in a point of R 8 , 3 . Let this be chosen as the line x.,=o and let x { ^=o be the line joining the two points on x. i =o and x 2 =o (other than their point of intersection) which belong to R K , 3 . The points 3, 7, 19 of x z =o and 6, 11, 15 of x 2 =o are then points of F 8 , 3 . Through each of the points 3, 7, 19 pass 3 lines of F 8 , 3 which contain 7 points of F 8 , :! . Since the one other point determines but one line with a given point it follows that through each of the points 3, 1, 19 passes one and but one line which contains no other point of F 8 , 3 . These 3 lines must meet x. 2 =o in a point of R 8 , 3 and hence all pass through the point 12 the intersection of x. 2 =o and * 1 =o. Any line through 4 ( i,o,o) other chan x 2 =o and x 3 =o intersects these three lines in points of R 8 , 3 and hence can contain no points of F 8 , 3 except as points of intersection with at,=<7. Therefore the other two points of F s , 3 lie on x t —0. But we know that the 9 points other than vertices on the sides of any triangle in PG(2,2-) are collinear by threes on 12 lines of which 4 pass through each point. Accordingly, if any one of the points 0, 10, 18 on x x =o be omitted there remain 8 points collinear by threes on 8 lines. Hence * The group G216 was studied at length by Maschke, Math. Annulen, Vol. 33 (1890), pp. 324-330, and the geometric properties of the group and its subgroups by Newson, Kansas Uni- versity Quarterly, Vol. II, No. 6 (Apr. 1901), pp. 13-22. 28 U. G. Mitchell: Geometry and F 8 , 3 must be F 9 , 3 with one point and the 4 lines through that point omitted and its group is the subgroup of F 9 , 3 G 24 of order 24 which leaves invariant a single point. < e ) pf — Q-r •=F 7 , 3 . Let / be any line of F 7 , 3 and P, Q, R the three points on / belonging to F 7 , 3 . Then there are but four other points A,B,C,D f F 7 , s not on / and these four points must lie two by two on two lines through each of the points of the complete quadrilateral of the other four points A,B.C,D. Let A,B,C, be taken (See Fig. 1, p. 4) as the vertices (7,0,0), (0,0,1) (0,1,0) respectively of the triangle of reference and D as the point (/,/,/). This deter- mines the coordinates of P,Q,R as ( 1,0,1), (0,1, 1), (1,1,0) respectively. Since these coordinates are in the GF(2) F 7 , 3 coincides with PG(2,2) and we can de- termine the number of collineations of each type by substituting n=l in the formulae given on p. 10, noting that type \ x of PG(2,2)is type I 3 of PG(2,2 2 ). This gives 56 collineations of type I 3 (type \ x in PG(2,2) ), 48 of type I of period 7, 42 of type III, and 21 elations, which, together with the identical trans- formation make a group G 1G8 of order 168. Every collineation in the group is of determinant unity and since it is of degree 7 and order 168 it is recognized as the simple group G ]68 first derived by Klein by the consideration of the trans- formation of the seventh order of elliptic functions.* Every group which leaves F 7 , 3 invariant must be a subgroup of G 1G8 and every subgroup of the G ]G8 leaves F 7 , 3 invariant! ( f ) r«-^ (g) 6 5 I =F • 2 15 c ' 2 ' 6 4 2 12 6 2 1 2 • (h) 6 3 F '. o _F ' °' 2 ' 2 9 6 ' 2 (i) =F Since the six points of F 6 , 2 are no three collinear we may choose any three of them for the vertices 4 (1,0,0), 12 (0,0,1), 17 (0,7,0) of the triangle of reference and any point not collinear with any two of these, say (l,I,i) may be taken as the fourth point. Since in PG(2,2 2 ) the choice of four points no three of which are col- linear determines uniquely (Cf. Corollaries 4 and 5 of Theorem 4) the other two points not collinear with any two of them, the fifth and sixth points are neces- sarily 13 (i,i,i) and 20 (ijj). Six points, no three of which are collinear, de- * Math. Annalen, Vol. 14 (1878), p. 438. Jordan, in determining the finite ternary groups missed both this group and the simple G36O. The G168. is discussed at some length by Burnside, Theory of Groups, pp. 208-209 and 302-305, and in Klein-Fricke's Modulfunctionen. ■f For a list of all groups whose degree does not exceed 8, see Miller, Amer. Journal of Math, Vol.21 (1899), p. 326. The types of substitutions and of subgroups of the G168 are given by Gor- dan, Math. Annalen, Vol. 25, (1885), p. 462. COLLINEATION GROUPS OF PG(2,2 2 ). 29 termine 15 distinct lines. Hence every collineation which leaves invariant the six-point also leaves invariant the associated fifteen-line. From this is follows that the groups leaving F,,,/, F„, 2 ", F ( „ 2 '" , invariant must either be the group leaving F , 2 invariant or subgroups of it. An elation E leaving F c , 2 invariant can not have more than two points of F „ on its axis, and since an elation interchanges by pairs all points not on its axis K must interchange by pairs at least four points of F,„ 2 . Since any four points no three of which are collinear can be transformed into any four such points by a collineation there exist in PG(2,2-) collineations interchanging by pairs any four points of F tl , 2 . All such collineations are elations because no other transforma- tions in PG(2,2 2 ) are of period two. Moreover, such an elation E leaves inva- riant each diagonal point of the complete quadrangle of the four points chosen and therefore the other two points on the diagonal line which are not diagonal points. These last two points must be the other two points of F„, 2 (Cf. proof of Cor. 5, Theorem 4). Since four points no three of which are collinear can be interchanged by pairs in three different ways it follows that there are three ela- tions leaving F , 2 invariant for each distinct quadrangle that can be chosen from F , 2 . There are then 3(0' , ;*>'4*3/4!)=4f> elations which leave F„ 2 invariant. Since any four points no three of which arc collinear can be transformed into any four such points by a collineation there exist in PG(2,2-') collineations per- muting any four points of F„, 2 in any given cyclic order. Such a transformation must be of period four and therefore of type III. A collineation T of type III which permutes in cyclic order any four points of F,., 2 must leave invariant one of the diagonal points of the complete quadrangle of the four points and inter- change the other two diagonal points. Since any two points of F„, 2 lie on the diagonal line of the complete quadrangle of the other four points but are not diagonal points it follows that if T permutes in cyclic order four points of F„,, it interchanges the other two points of F a>2 . Since any four points, no three of which are collinear, can be permuted in six different cyclic orders it follows that there are 6 (6*5 4 -3/4!) =90 collineations of type III which leave F„. 2 invariant. Since a collineation of type l x of period 5 leaves invariant one real point if it leave F fl , 2 invariant its center must be a point of F (! , 2 . The other points of F e , a form a conic of which the sixth point (the center) is the outside point. If 4 (i,o,o) be taken as the center and the other five points permuted in the order (1 13 20 12 17) the transformation is found to be T: px./=rx.,-\-x :i of type Ij and period 5. It was shown in § 3 that there are six different pairs of imaginary points on the axis of T determining six transformations of type I, with the same center and axis and no one a power of another. These correspond to the six different cyclic orders in which five points of the conic can be permuted and there are, therefore, G independent transformations of period 5 (24 altogether) 30 U. G. Mitchell: Geometry and leaving F G , 2 invariant and having the point 4 for center. Hence there are alto- gether G -24=144 collineations of type I ± leaving F G , 2 invariant. Since a transformation of type I 3 permutes in cyclic order any set of three points so related to the vertices of its invariant triangle that the six points are no three collinear, any collineation of type I 3 which has three points of F 6 , 2 for vertices of its invariant triangle must leave F , 2 invariant. Twenty distinct triangles can be chosen from the six points of F , 2 and hence 40 collineations of type I 3 leave F , 2 invariant in this way. We know, however, that each such collineation per- mutes the vertices of two triangles not in F r> , 2 either of which may be taken as the invariant triangle of a transformation of type I 3 which permutes the vertices of the two triangles in F p , 2 . For each pair of triangles that can be selected in F c , 2 there are then four transformations of type I 3 having invariant triangle not in F G , 2 which leave F , 2 invariant. Since ten pairs of triangles can be chosen in F G , 2 Lhere are 40 collineations of type I 3 which leave F c , 2 invariant in this way. It has been shown that the group which leaves F G , 2 invariant must contain as many as 360 collineations. Since the group can be represented as a substitu- tion group on 6 symbols it must therefore be either the alternating or symmetric group of degree six. But since there is no collineation which holds four of the points of F 6 , 2 each fixed and interchanges the other two the group can not be the symmetric group. The group which leaves F , 2 invariant is therefore the alternat- ing group on six symbols, shown by Wiman* to be identical abstractly with the the finite ternary group G 300 first set up bv H. Valentiner.f The group G 3fi0 is here characterized not only as a group on six points but since any one of the points can be taken as the outside point of the conic deter- mined by the other five points as a group leaving invariant a system of six conies. It is clear that every subgroup of G 3(i0 leaves F,., 2 invariant and every group in PG(2,2 2 ) which leaves F r> , 2 invariant must be a subgroup of G 3fi0 . 0) 5 4 2 10 =F„ 2 ; (k) 5 2 2 5 F n ,/. Since the five points of F-, 2 are no three collinear they form a conic and the group which leaves F a , 2 invariant is therefore simply isomorphic with the group cf all transformations of points on a line. The group accordingly contains 15 elations, 20 type I 3 's and 24 type I/s of period 5 (Cf. § 3). Since every trans- formation which leaves a conic invariant must leave its outside point invariant this group is recognized as the subgroup of the G 3no which leave? a single point fixed. It is here represented joth as the group which leaves invariant a conic and the alternating group on five symbols (the five points of F-, 2 ). The group which leaves F 5 , 2 ' invariant must be the subgroup of the G„ which leaves the 10 lines of * Math. Annalen, Vol. 47, (1896), p. 531. f Kjoeb. Skr. (5) 5 (1889), p. 64. See Ency. d. Math., ffiss., Vol. I, p. 529. In deter- mining the finite ternary groups, Valentiner, who was apparently unaware of the previous work of Klein and Jordan, missed the G36O. COLLINEATION GROUPS OF PG(2,2»). U F B>a invariant in two systems of five lines each. It is therefore a group G 10 of order 10. (I) pr"M_F . < m >j i ,|_ F , |_2 0J ^4 ' l ■ ,, LL_LL ' The configuration F 4 , 2 is a complete quadrangle and since four points can be permuted among themselves in all ways by transformations in the plane the group must be the symmetric group G 21 of all transformations on the four points. The configuration F 4 , 3 ' is a simple quadrangle and hence its group is the subgroup G c of the G,,. <»> rt-r I 2 3 =F 3 , 2 . The configuration F„ a is the triangle and hence every transformation leaving it invariant must do so in some one of the following three ways: (1) Leave the three vertices each fixed; (2) leave one vertex fixed and interchange the other two; (3) permute the three vertices in cyclic order. These may be written down at once as follows: Under (1) there are 2 type I 3 's and G homologies; under (2) there are 18 type II's and 9 elalions; under (3) there are (> type I.,'s and 12 type I,,'s of period 3. Including the identity, then, there are 54 collineations in the group. § 8. Subgroups of the Group G 2880 Which Leaves a Line Invariant. All groups which leave invariant a set of collinear points must also leave in- variant the line / which contains the points. Since any four lines no three of which are concurrent can be transformed into any four such lines by a projective trans- formation in PG(2,p n ) the order of the group leaving a line fixed is N= (/,=«+/," )(^»)^"— 2/>"4-l)=/r"(A J "— 1 )(/>"— 1) For PG(2,2 2 ) this gives N—2880 and accordingly the group will be desig- nated as G 2880 . Since any line in the plane can be transformed into any other line in the plane by a collineation within the G 0048o it follows that G, ; „ 1H „ contains 21 conjugate groups G 2880 . Subgroups of G.. 880 which lean- a point not on I invariant. In determining the subgroups of G 2880 we shall first determine all subgroups which leave invariant at least one point not on / and then all subgroups which leave invariant no point not on «\ Taking / as the line * 3 =<> (or the line at infinity) every collineation in the G 2(n is of the form x'=a l x-\-a.,y-\-a l! y'—btx+bj+b where x and y are nonhomogeneous point coordinates. Selecting the point not on / as tie origin every collineation in the G., s „„ which leaves it invariant is of the form x'=a l x-{-a.,y 1 < : y '=b x x+b,y But it has been seen in § 3 that in homogeneous coordinates the group of all trans- 32 U. G. Mitchell: Geometry and formations of the form T is the group G eo of all transformations of points on .1 line. Since G co is the alternating group on five symbols it contains the follow- ing subgroups:* I. Subgroup leaving all points of the line fixed. 1 self-conjugate G x — the identity. T l : #'=«!#, y'-.=a 1 y. II. Subgroups leaving invariant two points of the line. a. Those leaving each point of the pair fixed. 10 groups G 3 each conjugate to the G 3 of transformations of the form T 2 :x , =a 1 x, y , =b 2 y i a y b., in the GF(2-). b. Those leaving the pair invariant. 10 groups G 6 each conjugate to the G 6 of transformations of the form T subject to the restriction that either a 1 =b 2 =o or a.,=b 1 =o. 10 groups G 2 each conjugate to the subgroup of G 6 for which the coeffi- cients are in the GF(2). III. Subgroups leaving one point of the line invariant. 15 groups G 2 each conjugate to the G 2 of transformations of the form T 3 : x'=a 1 x-\-a 2 y J y'= a x y, where tf^^are in the GF(2 2 ). 5 groups G 4 each conjugate to the G 4 of transformations of the form T 3 where a lt a 2 are in the GF(2 2 ). 5 groups G 12 each conjugate to the G 12 of all transformations of the form T 4 : x'=a 1 x-\- a 2 y, y'=b s y, where a 1 ,a 2 b 2 are in the GF(2 2 ). IV. Subgroups leaving invariant a pair of imaginary points on the line. a. Those leaving each point of the pair fixed. 6 groups G r> each conjugate to the G- of all transformations of the form T subject to the condition that a 1 --]-ia 2 2 -{-i 2 b 1 2 -{-b 2 2 -\-a 1 a 2 -\-i'-a 1 b 1 -\-ra 2 b 1 -\-a 2 b 2 -\-rb 1 b 2 =o b. Those leaving the pair invariant. 6 groups G 10 each conjugate to the G 10 of all transformations of the form T subject to the condition that If T be taken as a transformation in nonhomogeneous coordinates we have a group G 60 simply isomorphic with G 60 5 or a group G 180 triply isomorphic with G 00 5 according as the determinant of the group of transformations of the form T 3 is unity or unrestricted within the GF(2 2 ). Corresponding to the above groups on the line there are then the following groups in the plane which leave invariant the line / and point (o,o). I. Subgroups leaving all points of / fixed. 1. Of determinant unity, 1 self-conjugate G v 2. Of determinant not restricted, 1 self -conjugate G 3 . II. Subgroups leaving invariant a pair of points on /. 1. Those leaving each point of the pair fixed. a. Of determinant unity, 10 conjugate G 3 each leaving a triangle invariant. b. Of determinant not restricted, 10 conjugate G 9 each leaving a triangle in- variant. All groups of degree less than 6 were obtained by Serret. COLLINEATION GROUPS OF PG(2,2 2 ). 3J{ 2. Those leaving the pair of points invariant. a. Of determinant unity, 10 conjugate G„ each leaving a triangle invariant. b. Of determinant unrestricted, 10 conjugate G 18 each leaving a triangle invariant. III.. Subgroups leaving one point of / fixed. 1. Of determinant unity, 15 conjugate G 2 each leaving invariant a point of lines. 5 conjugate G« each leaving invariant a point of lines. 5 conjugate G J2 each leaving invariant the line /, the Ar-axis and the origin IV. Subgroups leaving invariant a pair of imaginary points on /. 1. Subgroups leaving each point of the pair fixed. 6 conjugate G„ of determinant unity. 6 conjugate G ]3 of determinant not restricted. 2. Subgroups leaving the pair invariant. 6 conjugate G 10 of determinant unity. 6 conjugate G 30 of determinant not restricted. Subgroups of G JSSII which leave no point not on I invariant. In the discussion which follows the term translation will be used to indicate an elation having the line / for axis and the term elation will be used only for an elation whose axis is not /. In determining the subgroups of G., 880 which leave invariant no point not on / we shall first determine all such subgroups containing no translations and then all such subgroups containing translations. Let G n be a subgroup of G, 880 which leaves no point not on / invariant and contains no translation. If G n contain an homology H having / for axis and A for center, G„ must contain at least one transform H' of H having some other point than A for center. One of the products H'H or H'H- is of determinant unity and leaves all points on / fixed. It is therefore a translation. Hence G n can contain no collineation other than the identity leaving all points of / fixed and, consequently, every such group must be simply isomorphic with the G 00 . We have seen (ante p. 30) that there is a group G, !0 leaving invariant a point conic and its outside point. By duality there is a G« leaving invariant a line conic and its outside line. Since the line / is the outside line of 48 different line conies there are 48 such groups G 60 which leave / invariant. Every transformation of the form *'=* + * is a translation and the group of all transformations of the form E where a and b are marks of the GF(2-) is a G 10 leaving every point on / fixed. Unless a=b=o E is of period 2 and hence G ]6 contains 15 cyclic subgroups of order two. If a=o and b be allowed to take on all values in the GF(2 S ) or if b=o and a be in the GF(2 2 ) a group of order 4 is obtained, consisting of all translations leaving fixed all lines through a given point P on /. Such a group will be desig- nated as a G 4 (P). If a be allowed to take the value o and but one other value and b be restricted in the same way a group of order 4 is obtained containing be- sides the identity 3 translations no two of which have the same center. Since such a group leaves invariant a complete quadrangle of which the centers of the three elations are the diagonal points, it will he designated as a G,(Q). If a be in the 34 U. G. Mitchell: Geometry and GF(2 2 ) and b in the GF(2) a group G 8 is obtained leaving invariant a point P on / and interchanging the four lines other than / through P by pairs in a given manner. The G 16 is an Abelian (or commutative) group since if Ei and Ej be any two clations in G 18 E j E i =E 1 E j . Consequently E j E 1 E j - 1 =E i E j E j - 1 =E 1 , and every subgroup of G 16 is self-conjugate within the G 1C . Also every two translations and their product form (with the identity) a group G 4 , for if EiEj=E k , E 1 =E k E j =EjE k and E j =E k E 1 =E i E k . Hence G 10 contains 15-14/0=35 sub- groups G 4 .* We shall next determine the subgroups of G 2880 which are such that every col- lineation in the group either leaves invariant a point not on / or is the product of such a collineation and a translation in the group. * It may be of interest to note that the Gi6 can be represented as a three-space PG<3,2) by letting the G2, G4, G$, correspond to the points, lines and planes, respectively, of the three-space. The three-space S3 has The Group Gi6 has 15 points, 3 5 lines, 15 planes, 15 subgroups G2, 35G4, 15 Gf, arranged arranged 3 points on each line, 3 G2 in each G4, 7 points on each plane, 7 G2 in each Gs, 7 lines through each point, 7 G4 containing each G2, 7 lines in each plane, 7 G4 contained in each Gg, 3 planes through each line, 3 G« containing eachG4, 7 planes through each point. 7 Gs containing each G2. If the three-space S3 be represented by the notation for a configuration (Cf. Moore, American Journal oj Mathematics, Vol. 18, pp. 264-303; Veblen and Young, Projective Geometry. Vol. I, p 38), the same table exhibits the structure of the group Gi6. In the table, S is a point, Si a line, S 2 a plane, and the interpretation is obvious from the parallelism given above. COLLINEATIOX GROUPS OF PG(2,2 8 ). 35 The translations in any subgroup of G 2880 form a self -con jugate subgroup G k . If any group G„ has a system of transitivity S which is also a system of transitivity of its self-conjugate subgroup G k of translations then every collineation in G„ is either a collineation leaving a point O of S invariant or the product of such a col- lineation and a translation ; for, let O be taken as the origin and let T be any collineation in G„. If T displaces O it changes O to some point A in S. But since S is a system of transitivity for G k there is in G n a translation T t changing A to O. Hence T 1 T=T 2 a collineation in G n leaving () invariant. From T 1 T=T, we have T=T 1 T 2 . Every such group G n can be obtained then by extending the groups leaving a point fixed by means of translations. In determining the groups below, S will be used to indicate the system of transitivity common to the group obtained and the extending group of translations. E,, E 2 and E will be used to indicate the forms of translations as follows: l ~ y'=y ^ /=y+* y'=y-H The product of T = ^T'^V and E is of ^ form y =b lX +b 2 y >Y T? =r -x'=•'=>•+<*•_.• Since TST"'=S we have a 1 ==i, a.>=0 and every transformation in the G„ is of the form T,: x'=x-\-ay-\-b, y'=cy-\-d. If r=i T, is of period 2 if cd=o and of period 4 if ad=i. If c=i or r and b=acd, T, is of period 3., If r=i or r and b is not equal to aid, T is of period (>. Hence every transformation in G„ is of type II, III, IV or V. Also, all transformations of types III and V have for center the point 4=(//m/) which is the center of S. all homologies have for axis some line other than / through 4 and for center some point ottier than 4 on /, and all transformations of type II have 4 as the point of intersection 38 U. G. Mitchell: Geometry and of the two invariant lines. Since no homology has / for axis, there are but two transformations (S and the identity) in G n which leave / point-wise invariant. G n is, therefore, at most (2, 1) isomorphic with some group in one dimension leaving a point on the line invariant and its possible orders are (cf. p. 32) 4, 8, 12 and 24. That G n can be a cyclic G 4 follows from the fact that every system of transitivity of the G 2 contains but two points. G n can not be a G 4 whose transformations are all of period 2, since if E be one of the elations in such a G 4 , the G 2 and the G 4 have the same systems of transitivity on -the axis of E. If G u be of order 8 and contain trans- formations of period 2 only it must contain two elations, E x and E 2 , having the same axis l t , since such a G 8 must contain six elations and there are but four lines other than / through 4. We then have E 1 E 2 =E 3 a third elation having 1^, for axis. Since E 1? E 2 and E 3 have the same center and axis they, together with the identity, form a group G 4 (P). The other three elations in the G 8 would be SE n SE 2 and SE 3 ; but since each of these elations has the same systems of tran- sitivity on l x as S, G„ can not be such a G 8 . Accordingly, if G n be of order 8 it must contain a cyclic subgroup G 4 consisting of the powers of a transformation U of type III and since U must leave / invariant we have U"^S. Hence U is< of the form U: x'—x-\-ay-\-b, y'=y-\-a' 2 where a is not zero. If the G 8 contain an elation it must be of the form E: x'=x-\-a 1 y-\-b 1J y'=y where a x is not zero. If a=a x the product EU: x'=x^\-b-\-b x -\-i, y'=y-\-a 2 is a translation different from S. If a is not equal to a { the product EU: x f =x-\-(a-\-a 1 )y J r b-\-b 1 -\-a 1 a 2 , y'=y-\-a 2 is a transformation of type III whose square must be identical with S. If (EU) 2 : x'=x-\- a x a 2 -\-i, y'=y be identical with S we must have a Y (i 1 -\-i=i or a 1 a' 2 -\-i=o; but if tf 1 tf 2 -|-l=l, a 1 a 2 =0 which is impossible since neither a nor a x can be zero, and if a 1 a 2 -\~i=0, a x d 2 =i which is impossible since a and a x are different marks of GF(2 2 ). Hence if G n be of order 8 it can contain no elations and must have 3 cyclic subgroups of order 4. Taking U x : x'=x-\-a 1 y-\-b l , y'"» y-f-tf, 2 as any other transformation of period 4 than U or U 3 in the G 8 we have UU X : x'=x-\-{a-\-a 1 )y-\-aa 2 -\-b-\-b^ y'=v+fl 2 +,, b., in the G m no matter what marks of the GF(2) a' and b' may be it follows that a it a 2 , £,, b. z are in the GF(2) and even- transformation in the G m is of determinant unity. Consequently, G m can contain no homology and is at most (4, 1) isomorphic with some group on the line leaving invariant a pair of points. The possible orders of G m are, therefore, (cf. p. 32) 12 and 24. The G 4 (Q) leaves each point on / (x^—o) invariant and permutes the other 36 points in four systems of transitivity each consisting of four points no three of which are collinear. These four quadrangles are Q,==(0 16 20 18) Q 2 s=(l 14 8 5) ; Q 2 =(2 13 6 15) ; Q 4 =( 9 10 12 11) (cf. Table of alignment, p. 3). Every transformation in G m is of the form T: x'=a 1 x-\-b 1 y-\-c l , y'= n.je-\-b z y-\-c i where a x , a 2 , b it b 2 are in the GF(2), and must, therefore, be of type V (elation, period 2), type I a (period 3) or type III (period 4). From the forms of T 2 , T : \ T 4 it appears that every elation in G m must be of the form E, : x'—y-\-k, y'=x-\-k, or of the form E.: x'=x-\-y-\-c, y'=y, or of the form E 3 : x'=x, y'=x-j-y-\-d ; every transformation of type I, must be of the form T, : x'=y-\-m, y'=x^-y-\-n, or of the form T 2 : x'=x-|-y-j-r, y'—x-\-s; every transformation of type III must be of the form U, : x'=y-\- k, y'=x-\-l where / and k are not the same mark, or of the form IL: x'=x-\-y-\-c l , y'— y-\-c t% or of the form U„: x'=x-\-d ly y'=x-\-y-\-d 2 . If G m be of order 12 it must be the direct product of a cyclic G.,(cvc. I 8 ) and the G,(Q). But each transformation of period 3 in G m must leave one of the quadrangles Qi(i=l, 2, 3, 4) invariant and permute the other three in cyclic order. Hence some one of these quadrangles would be a system of transitivity of the (i,_. generated by any G a (cyc I 3 ) and the G 4 (Q) and G m can not be or order 12. If G m be of order 24 it must contain a 40 U. G. Mitchell: Geometry and transformation T of period 3 leaving invariant a quadrangle Q a (a=l, 3, 3 or 4). Every transformation of the form T 1 or T 2 is seen to leave invariant the two points 7==(i 1 i,o) and 19^(/,j,0) on /, and since every translation in the G 4 (Q) leaves every point on / invariant the eight products obtained by multiplying T and T 2 by the transformations in the G 4 (Q) are eight transformations of period 3 leaving Q a invariant. Moreover, G 24 can not contain any transformation T t of period 3 not included among these eight, for the products of T and the three other transformations leaving Q a invariant and making the same permutation of points on / as T give the three translations in the G 4 (Q) and consequently one of the products TjT or TjT 2 would be a translation not in the G 4 (Q). Hence, the G 4 must contain, besides the G 4 (Q), 8 transformations of type I. { leaving Q a invariant and some transformation S a of period 2 or 4 transforming Q a into one of the other quadrangles. Let S } be of period 2 or 4 interchanging the four quad- rangles in any order R 1 =(Q a Q b ) (QcQd) where a, b, c, d are the numbers 1, 2, 3, 4 in an arbitrary order. Since half of the transformations of period B in the G 24 must make the transformation R 2 =(Q a ) (QbQcQd) on the four quad- rangles Qi(/=1, 2, 3, 4) the G 24 would then contain a transformation of period 3 making on Qt(t— 1, 2, 3, 4) the transformation R 2 R,= (Q a QbQc) (Qd) which has just been shown to be impossible. Hence the G 24 can contain no transformation of period 2 or 4 interchanging the Q t (z'=l, 2, 3, 4) in pairs. Also, G 24 can not contain a transformation of period 4 permuting the Qi (*=1, 2, 3, 4) in cyclic order for its square would be a transformation of period 2 interchanging them by pairs. Hence, the G„ 4 must contain a transformation S x of period 2 or 4 which transforms the Q, (i— 1, 2, 3, 4) in the order R 3 =(Q a Q b ) (Q c ) (Q d ) ; but since R 2 R 8 =(Q a Q b Q c Q d ) the G 24 would then contain transformations permut- ing the Qi(zWl, 2, 3, 4) in cyclic order which has just been shown to be impossible. Hence G 2880 contains no subgroup G m having a G 4 (Q) as its largest self-conju- gate subgroup of translations and such that the G 4 (Q) has no system of transi- tivity which is also a system of transitivity of the G m . Subgroups containing a self-conjugate G 4 (P). Let G k be a subgroup of G 2SP „ having a G 4 (P) as its largest self-conjugate subgroup of translations and such that the G 4 (P) has no system of transitivity which is also a system of transitivity of the G k . Selecting the G 4 (P) as the group of all transformations of the form S: x'=x-\-a, y'=y, and T: x'=a 1 x-\-b l y-\-c 1 , y f =a 2 x-\-b 2 y-\-c 2 as the general transformation in the G 2880 we have TST" 1 : x / =x-\-a 1 a, y'=y-\-aa.,. Hence in order that TST" 1 may belong to the G 4 (P) we must have a 2 =o and if we also have «,=/ each translation in the G 4 (P) is self- conjugate. Hence every transformation in G k is of the form T t : x f =a i x-\-b 1 y-\-c l , y'=b.,y-{-c 2 . The G 4 (P) consists of all transla- tions having / for axis and 4 =(1,0,0) for center. The G 4 (P) has four svstems of transitivity, ^=(0 1 16 14), / 2 =3(8 5 18 20), / 3 =(2 13 10 9), / 4 ss(16 16 12 11) and in each system the four points are collinear. From the form of T, 2 it appears that every elation in G k is of the form E: x'=x-\-b 1 y-\-c 1 , y'=y (where b } is not zero) and, consequently, has 4 for center and leaves each ?i(i«= =1, 2, 3, 4) invar- iant. Hence there is no G k containing translations and elations only. From the form of T/ it appears that every transformation of type III in G k is of the form U: x'=x-j-b 1 y-\-c i , v'=y+ f 2' where neither b 1 nor c 2 is zero. The 12 trans- formations for which c 2 =i make the interchange {lj 2 ) (/ 3 / 4 ) ; the 12 for which r 2 =/' make the interchange (IJ^'ilJJ ; and the 12 for which c 2 =r make the inter- change (/j/J (/ 3 / 2 ). Since the G k can not leave any ? t (i— 1, 2, 3, 4) invariant it COLLINEATION GrOUI'S OF PG(2,2 2 ). J] must either be transitive on the /, or interchange them by pairs. Suppose G k to make the interchange (/,/.,) (/.,/ 4 ). Such a G K can not contain a homology having / for axis, for the homology would permute three of the lines /j in cylic order. Hence, the G k would be at most (4, 1) isomorphic with some group on the line leaving a point invariant and its possible orders would be (cf. p. 32) 8, 12, 1<>, 24, 48. Since G k is transitive on 8 points its order must be divisible by 8 and can not be 12. If G k be of order 8 and make the interchange (/,/.,) (/.,/ 4 ) it must contain a transformation of the form U, : x'=^=x-\-b l y-\-c i , y'=y-|-/. The products of a U, and the transformations in the G 4 (P) are four transformations of the form U, where Z>, is fixed and r, is any mark of the GF(2-). These 4 transformations of type III and the transforma- tions in the G 4 (P) form a group G s which is a G k . Hence there are three such subgroups G s (one for each value of b x ) interchanging the /i(i=l, 2, 'i, 4) in the order (/,/.,) (/ :i / 4 ) and similarly 3 subgroups G s for each of the orders (/,/..) (A./;) and (lj 4 ) (IJ.,)- No G k can contain more than 8 transformations of the form U,, since the product of \J l : x'=x-\-y-\-c l , v'=y-\- \J / : x , =x-\-iy J rf i , y'=y+ /, and \J /' : x'=x+ry+c u y'=y+ 1, is U, U/ U.": *'=*+<-,+», y'=- y-\-r, which is a translation not in the G 4 (P). The product of any two transfor- mations of the form U, and U,' is U 1 U/=E 1 : x'=x-\-i-y-\- 1 , y'=y which is an elation having 4 for center. The products of E 4 and the four transformations in the G.f(P) give all elations of the form E,': x'=x-\-i-y-\-c, y'—y. The product of any two elations of the form E/ is a translation in the G 4 (P). Also all prod- ucts of transformations of the form E/ with transformations of the form \J l or U,' are of the form U,' or U, respectively. Hence, the 12 transformations of the forms U,, U/ and E,' together with the G 4 (P) form a Group G,„ which is a G k . Obviously there may be two other such groups G 1(5 making the interchange Ci OCa O and similarly •'* groups G,„ making the interchange (,/, l 3 )(l a /,) and 3 groups G M! making the interchange (/, l 4 )(l- 2 / :t ). Also, since the product of an elation of the form E: x'=x-\-by-\-c, y'=y and a transformation of type III of the form U: x'=x-\-by-\-c u y'=y-\-c, is EU : x , =x-\-c 1 -\- bc 2 -{-c,y'=y-\-c.,, which is a translation not in the G,( P), there is no other type of group of order lfi which can be a G k interchanging the l\ by pairs. A G k of order 24 or 4S which makes the interchange (/,/..)(/.,/,) must con- tain transformations of period 'J. Every such transformation would leave each /i invariant and hence would be a homology H having 4 for center. Such a G k must also contain some transformation T making the interchange (/, /;>)(/., / 4 ). But T can not be of type II, for in that case T 3 would be a translation not in the G 4 (P), and T can not be of type III, for in that case HT would be such a trans- formation of type II. Hence there is no G k of order 24 or 48 interchanging the l\ by pairs.. No G k can contain a homology H having / for axis; for, if P, be the center of H, G k must contain some transformation T transforming P, to some point P/ which is not collinear with P, and 4. Since H can not leave P,' invariant HT and TH transform P, to different positions and. hence, H and T are not commu- tative THT' , =^-H, is, therefore, a homology having P, for center and one of the products HH, or H 2 H, is a translation not in the G 4 (P) since it would have for center the point where the line P,P/ cut /. Accordingly G k is at most (4, 1) isomorphic with some group on the line leaving a point invariant and its possible orders are 8, 12, 16, 24 or 48. If G k be transitive on the /, it is transitive on all points not on / and its order must, therefore, be divisible by 1<>. Consequently such a transitive G k must be of order 16 or 48. If such a G k be of 42 U. G. Mitchell: Geometry and order 16 it must contain two transformations of type III, U. : ;(•'=.*• -f-^iV+^n y'=y-\-c lt and U 2 : x'=x-\-a 2 y-\-b 2 , y'=y-\-c 2 where c, and c 2 are neither one zero and are distinct from each other. The product UjU 2 is of tihe form U 3 : x'=x-\-a 3 y-\-b 3 , y'=y-\-c 3i where a 3 is different from a x and a 2 and c 3 is different from c t and c 2 . Also a 1 must be distinct from a 2 , for otherwise U 3 is a translation not in the G 4 (P). All products U 2 U 3 are of the form \J 1 and all products 1-1,113 are of the form U,. Also, the square of each Uj(/=1, 2, 3) is a translation in the G 4 (P). Hence 12 transformations, 4 each of the forms U t , U 2 , U 3 , such that a x , a.,, a 3 are no two the same mark and c v c 2 , c 3 , are no two. the same mark, form, together with the G 4 (P) a group and the only type of group G, 6 which is a transitive G k of order 16. Since a x and c Y may each be chosen in 3 different ways and a 2 and c 2 may each then be chosen in 2 different ways thcrfc are 36 such groups G ]0 having the given G 4 (P) as its largest self-conjugate sub- group of translations. If G k be a transitive group of order 48 it must contain a transformation T of period 3. If T be a homology it must either have some line through 4 for axis or have 4 for center. That T can not be a homology, having / for axis has been shown above. That T can not be a homology having any other line through 4 for axis follows from the fact that if S be one of the translations in the G 4 (P) TS is of type II and (TS) 3 is a translation not in the G 4 (P). If T be a homology having 4 for center it leaves each / t ( z==l, 2, 3, 4) invariant. Since the G 48 must contain a subgroup G 16 which can have no transforma- tion other than identity in common with the cyclic G 3 generated by T, the group J G 16 , G 3 [ is the G 48 and leaves each /i(z=l, 2, 3, 4) invariant unless the G ]6 contain a transformation U of type III having 4 for center and having for axis a line through P 3 some point on / different from Pj and P 2 . Hence the G lS would contain at least 24 distinct homologies and since the products of these by U give 24 distinct transformations of type II the G 48 would contain more than 48 transformations. Hence a transitive G k of order 48 can not contain a homology. Since T can not be a homology it must be of type I 3 having for vertices of its invariant triangle a point A not on / and two points, 4 and some other point P r other than 4 on /. The cyclic G 3 generated by T together with the G 4 (P) gen- erates a G 12 consisting of all transformations of determinant unity leaving invari- ant the points 4 and F 1 and the line / a joining A to 4. But the G 48 must contain a subgroup Gi C having no transformation other than identity in common with the cyclic G 3 generated by T. Hence the G ]0 and the cyclic G 3 would generate a G 48 leaving / a invariant unless the G 16 contain a transformation U of type III inter- changing / a with some other line / b through 4. If the other two lines tihrough 4 be designated as l c and / d , U makes the transformation U=(/ a / b ) (/Jd) ^ n these lines and T makes the transformation T=(/ a ) (IbhU)- Hence TU= (IJJb)(U) is of type I 3 leaving / d invariant and UT= ( / a / b / d ) ( / c ) is of type 1^ leaving l c invariant. Also, the product of TU and UT is a transformation of type I 3 leaving the line / d invariant. Each of these transformations of type I 3 generates a cyclic G 3 , which, taken with the G 4 (P) generates a G 12 containing 8 transfor mations of type I 3 . These 32 transformations of type I 3 are such that each point not on / is an invariant point of two of them and each point on / (other than 4) is an invariant point of 8 of them. Hence the self-conjugate G 1G can not contain an elation E ; for, if T' be a transformation of type I 3 having an invariant point on the axis of E, ET' would be a transformation of type I 3 not among the 32 above named. The G 16 must, therefore, consist of 12 transformations of type III COLLINEATION GROUPS OF PG(2,2 S ). [3 and the G 4 (P). Taking the G 10 as all transformations of the form U : x'=x-\-ay-\-b> y'=y-\-a and the cylic G. t generated hy a transformation of type I 3 as all transfor- mations of the form T: x'=mx, y'=n , where mn—i and neither m nor n is unity, the products UT: x'=mx-\-any-\-b, y'=ny-\-a and TU : x'=mx-\-amy-\- readily verified that the product of any two transformations of the form U, UT and TU is of the form U, UT or TU. Hence they form a group G 48 which is a transitive G k . Subgroups containing a self-conjugate G s . There remains to be determined every subgroup G t of G 2880 which leaves fixed no point not on /, has a G 8 as its largest self-conjugate subgroup of translations and has no system of transitivity which is also a system of transitivity of the G 8 . A G 8 of translations consists of a G 4 (P) (all translations of which have for center the same point P on /) and 4 other translations each having a different center from any other in the G 8 . The G s leaves invariant besides / and P a pair of lines through P. Hence the G 8 has two systems of transitivity of 8 points each and a G t must be transitive on the 16 points not on /. No G t can contain a homology H having / for axis ; for, if T be a trans- lation in the G 8 having a point P' different from P for center H and T are not commutative (since HT and TH transform the center of H to different positions) and HTH" 1 is a translation having P' for center and not in the G 8 . Hence, a G t contains no transformation leaving / pointwise invariant except the translations in the G 8 and is at most (8, 1) isomorphic with some group leaving invariant a point on the line. Its possible orders are, therefore, (cf. p. 32) 16, 32, 48 and 90. Tak- ing P as the point 4=(/,a,o) the G 4 (P) in the G 8 becomes the four translations of the form S: x'=x-\-a,y'=y and the four other translations in the G„ may be taken as the four translations of the form Sj : x'=x-\-a, y'=y-|-/. Since a G t can contain no translation not in the G 8 a translation of the form S must be trans- formed into a translation of the form S, and a transformation of the form S, must be transformed into a translation of the form S t by every transformation in G t . It has already appeared above that the first of these conditions requires that every transformation in a G t shall be of the form T t : x'=a l x-\-b l y-\-c l , y'=b.,y-\-i... Transforming Sj through T 1 gives TS,^" 1 : x'=x-\-aa lt y'=y-\-b 2 and hence «,=--l, b. 1 =l and every transformation in G t is of the form T 2 ": x'—x-{-b i y-\-c l , y'=y-\-c... If b y =o or if c 2 =o T 2 is of period 2. Hence, every elation in a G t is of the form E: x'=x-\-b 1 y-\-c 1 ,y'=y, where b t is not zero. If neither b x nor c 2 is zero T, is of period 4. Accordingly a G t can contain only transformations of period 2 or 1 and must be of order 16 or 32. The two systems of transitivity of the G s are left invariant by every elation of the form E and hence a G L must contain a transformation U of type III. Under the G 8 of translations the pair of lines y=o, y«=i is one system of transitivity and the pair of lines y=i, y=r (equations in non-homogeneous coordinates) is the other system. Hence U must be of the form Uj : x f ==x-{-ay-\-b, y'=y-\-i or of the form U 2 : x'=x-\-ay-\-b, y f ==y-\-r- But the product of a translation of the form S, and a transformation of the form \J 1 or U 2 is of the form U s or U,, re sp ec ti vely, and hence every G t must contain transformations of both forms. A G t of order 16 must, therefore, contain, besides the Q H of translations, 4 transformations of the form U, and 4 of the form U 2 where a has the same value for all of these trans- formations of type III. Since every product U,U 2 and U 2 U, is a translation fn the G., these 16 transformations form a Group G„ which is a (i t . Since there are 3 choices for a there are 3 such groups G, fl having the given G s as the largest sub- 44 U. G. Mitchell: Geometry and group of translations. Also, it appears from the above that every G t must contain at least one such G 1C as a subgroup. This G ]6 will be taken as the subgroup con- sisting of the 8 translations in the G 8 , 4 transformations of the form U/: x'=x-\- y-\-c, y'=y-\-i and the four transformations of the form U/ : x'=x-)-y-\-c, j/=_y-|-r. For convenience this G 16 will be referred to as the group G'. A G t of order 32 must contain some transformation T of period 2 or 4 not in the subgroup of order 16 taken as G'. If T be of period 2 it must be of the form E: x / =x-\-ay J r b, y'=y. The products U/E: x'=x J r {a-\-i)y-[-b-\-c J y '=yJ r i > and U/E: x'=x-\-.(a-\-i)y-\-b-\-c, y'=y-\-i~ are translations not in th^ G 8 of translations if a=i. If et=*i, the G 32 must contain 4 transformations of type III of the form U/': x'=x-\-ry-\-m, y'=y+z, and 4 of the form U 2 ": x f =x-\-ry-\-m, y'=y-\-i'. Also, the products of E and the translations of the form S 1 introduce 4 transformations of the form U 3 : x f =-x-\ r iy-\-b, y'=y-\- r. Thus are determined, besides the 8 translations in the G 8 , 4 transformations of each of the forms E, U/, U," and U ;i . These are all of the form U: x'=x-{-ay-\-b, y / =-j.-|_ f where c=i or r if o=i or r and c=o or / if o — or i. Taking U t : x'=.x--j-tf 1 };-|-Z> 1 , y'=y-\-Ci as a second transformation of the form U the product is UU a : x'=x-\-(a 1 -\-a)y-\-ac 1 -{-b i -\-b, y'=y+f-)-r 1 . If o=o or a x =o or \i 0=0-1 it may be verified by inspection that this product is one of the given forms. For the other possibilities the following table gives the results: a <*i c Ci Resulting forms. I i r or i o or i U/' or U a " I V v or z r or i E or U 3 i i or I r or i U," or Uo" i P o or i i or i' 1 U/ or U/ r i i or r t or r E or U 3 r 1 i I i or r o or / U/ or U./ Hence these 32 transformations form a group G 32 which is a G t . If a in E had been chosen as r, a G 32 of the same type would have been dete* mined. Hence there are two groups G 32 having the G' as a subgroup. SUMMARY. 1. In the finite projective plane PG(2,2 n ) the diagonal points of a complete quadrangle are collinear. 2. If an outside point of a conic be defined as any point of intersection of tan- gents to the conic, every conic in the PG(2, 2 n ) has but one outside point and all tan- gents to the conic concur at that point. Through every point other than the outside point there passes one and but one tangent to the conic and every line through the out- side point of a conic is a tangent to the conic. 3. In the PG(2 > 2 n ) six and but six points can be chosen such that no three of the set are collinear. 4. All of the types of projective collineations of the ordinary projective plane are present in the PG(2,2 n ) and the number of such collineations in the PG(2,2 2 ) is 60480. COLLINEATIOX GROUPS OF PG(2,2 2 ). |.~» 5. Every subgroup of the group G 004ao of all projective collineations in the PG(2,2 2 ), except a self-conjugate G 20 j 60 leaves invariant a real figure [real with- in the PG(2 f 2-)] or an imaginary triangle. (). There are 8 kinds of groups leaving invariant an imaginary triangle and their list is given in Theorem 11. 7. All configurations in tlie PG(2,2 2 ) and the groups characterizing them are determined. These groups include the finite groups of the ordinary projective plane. Consequently, the simple G 360 , the Hessian (j L ,,,. and the simple G, M are all subgroups of the G e0480 and within the PG(2,2 2 ) the geometric invariant of each is a real configuration. 8 The subgroups of the G., MS „ which leaves a line invariant are chiefly (1,1) or (3,1) isomorphic with groups on the line, but certain groups of higher isomor- phism are present and are determined. THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW am INITIAL FlNE~OF 25 CENTS A . ^ASSESSED FOR FA.LURE TO RETURN E? r N o R to E J oo 50 o C N EN t T he° Seventh o*v OVERDUE. OCT 24 1932 17 1932 0#294$P \2fc' 3 Apr '49 FA NOV 3 1936 OCT 7 i2May'49JS AV Auf(W&* r - LU 21-50?n-8,'3-J 2880 61 UNIVERSITY OF CALIFORNIA LIBRARY >,ir *,fy