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Geometry of the Mathieu Groups and Golay Codes

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Geometry of the Mathieu Groups and Golay Codes Proc. Indian Acad. Sci. (Math. Sci.), Vol. 98, Nos 2 and 3, December 1988, pp. 153-177. 9 Printed in India. Geometry of the Mathieu groups and Golay codes ERIC A LORD Centre for Theoretical Studies and Department of Applied Mathematics, Indian Institute of Science, Bangalore 560012, India MS received 11 January 1988 Abstraet. A brief review is given of the linear fractional subgroups of the Mathieu groups. The main part of the paper then deals with the proj~x-~tiveinterpretation of the Golay codes; these codes are shown to describe Coxeter's configuration in PG(5, 3) and Todd's configuration in PG(11, 2) when interpreted projectively. We obtain two twelve-dimensional representations of M24. One is obtained as the coUineation group that permutes the twelve special points in PC,(11, 2); the other arises by interpreting geometrically the automorphism group of the binary Golay code. Both representations are reducible to eleven-dimensional representations of M24. Keywords. Geometry; Mathieu groups; Golay codes; Coxeter's configuration; hemi- icosahedron; octastigms; dodecastigms. 1. Introduction We shall first review some of the known properties of the Mathieu groups and their linear fractional subgroups. This is mainly a brief resum6 of the first two of Conway's 'Three Lectures on Exceptional Groups' [3] and will serve to establish the notation and underlying concepts of the subsequent sections. The six points of the projective line PL(5) can be labelled by the marks of GF(5) together with a sixth symbol oo defined to be the inverse of 0. The symbol set is f~ = {0 1 2 3 4 oo}. The group L2(5) of homographies on the line is generated by the modulo 5 transformations on fl: ~:i~i+1, fl:i--*--i, ),:i--*--l/i. (1.1) As permutations, ~=(01234) fl = (14)(23) (1.2) = (0 ~)(14). The group S 6 of all permutations of the six symbols is generated by a and ~, where =(0oo). (1.3) (As a modulo 5 transformation, z: i ~ 1/i3). Alternatively, the S 6 is generated by ct and lr, 153 154 Eric A Lord where = (0 ~)(14)(23) (1.4) (Tr:i~ -- 1/i3). The mapping 0 on S 6 defined by ~o = ~ ~o = rc (1.5) is an involutory outer automorphism of S 6. It is obviously not an inner automorphism because the fifteen transpositions correspond, under 0, to the fifteen regular involutions (products of three mutually commuting transpositions). These correspondences are the same as the correspondences between Sylvester's duads and synthemes of six symbols [11 [21 [6]. Any transposition appears in the structure of three of the regular involutions and each regular involution is comprised of three transpositions--a Cremona-Richmond 153. The self-dual property of the 153 corresponds to the involutory property of 0 (02 = 1). The S 6 can alternatively be generated by the five transpositions (0 oo), (1 oo) (2 oo) (3 oo), (4 oo) or by the five regular involutions rco = (oo 0)(14)(23), ~z1 = (oo 1)(20)(34), ~z2 = (oo 2)(31)(40), (1.6) rr 3 = (oo 3)(42)(01), rr4 = (oo 4)(03)(12), (obtained by acting on zr with the powers of 00. Under the action of 0, the generators ~, fl, y of L2(5 ) are mapped to ~, fl, 6, where 6 = (12)(34). (1.7) Whereas the original L2(5), generated by ~, fl, y is transitive on the six symbols, the L2(5 ) generated by ~, fl, 6 acts on only five of them. It is in fact the alternating group A 5 of even permutations of {0 1 2 3 4}. We have the isomorphism L2(5) "-, As. The action of0q fl, y on the set f~ induces the action of ~, fl, 6 on the set of five objects (1.6). (~i) ~ = n,~, (rri)# = ~i#, (zc~y = zi~ (1.8) (the notation here is 7r~ = ~t- 1~0~, etc). The four permutations ~, fl, y and 6 together generate the group A 6 of all even permutations of six symbols. Indeed, this A 6 is generated by ~, fl, and 6 only, since = (r6)2. The $6 together with its outer automorphism 0 generates a group $62 (a group with an invariant subgroup $6 of index 2). The eight points of the projective line PL(7) can be labelled by the set f~ = {G 1 2 3 4 5 6 ~} consisting of the marks of GF(7) and the extra symbol oo. The group L2(7) of homographies on the line is generated by the modulo 7 transformations ~:i~i+ 1, fl:i~2i, y:i~-- 1/i. (1.9) Mathieu groups and Golay codes 155 As permutations, a=(0123456), fl = (124)(365), ? = (0oo)(16)(23)(45). (1.10) Analogously to the case of PL(5), there exists another L2(7) that acts on only seven of the eight symbols; it is generated by e, fl and 6 = (12)(36). (1.1 t) The two groups L2(7) are mapped to each other by an involutory outer automorphism of the group F generated by e, fl, ~ and 6. In Figure 1, the vertices of Fano's configuration 73 have been labelled by the marks of GF(7) (observe that e cyclically permutes the vertices of the 'self-inscribed and circumscribed heptagon' 0123456, and that fl rotates the figure and 6 reflects it). The projective plane PG(2, 2) is a Fano 73 and 0~, fl and ~ generate its group of collineations--the simple group L3(2) of order 168. We have the isomorphism L2(7 ) ,,-L3(2 ). The quadratic residues of GF(7) are the elements of the set Q = {0124}. The complementary set is N = {356 oe }. Consider the seven pairs of four symbols obtained by operating with the powers of e on Q, N: Po= {0124}{356oo}, P1 = {1235}{460oo}, P2 = {2346}{501 oo}, P3 = {3450}{612oo}, (1.12) P,= {4561}{023 oo}, P5 = {5602}{134oo}, P6 = {6013}{245 oo}. The action of the L2(7) generated by e, fl, 7, on f~, induces the action of the Lz(7 ) generated e, fl, 6 on this set of seven objects; ~:Pi~Pi~, fl:P~Pi,, ?:Pi~Pi~ (1.13) 6- - -5 1 Figure 1. 156 Eric A Lord The seven objects Pi are the seven pairs of complementary tetrads of a Steiner system S(3, 4, 8). (Conway employs the involution rc = (az 0)(13)(26)(45) instead of our object Q, N. The effect is the same. Our seven objects Pi are more suitable for our later purposes). The twelve points of PL(ll) can be labelled by the symbol set f~ = {01234 5 6789 X oo}. The first eleven of these symbols are the marks of GF(11). The homography group L2(11) is generated by ~:i--+i+1, fl:i--+3i, 7:i~--l/i (1.14) (modulo 11), i.e. by the permutations c~=(0123456789X), fl=(13954)(8267X), (1.15) = (0 oo)(1X)(25)(37)(48)(69). The set Q = {013 9 54} is the set of quadratic residues. The non-zero quadratic residues are the powers of 3 modulo 11. The complementary set is N = {8 2 67 X oo}. Observe that fl leaves invariant these two hexads and Y interchanges them. The Mathieu group M12 acting on the twelve symbols is generated by =, fl, y, A, where A = (2X)(34)(59)(67). (1.16) As is well known, Mx2 is the automorphism group of the Steiner system S(5, 6, 12). The 66 complementary pairs of hexads are the images under the action of M~2 of the generic hexad pair Q, N. Observe that fl = (TA)6 so in fact M12 is generated by ~, y, A. Our original subgroup L2(11) was generated by ~, fl, ~. Another Lz(11) subgroup is generated by ~, fl, A. The original L2(11) is transitive on the 12 symbols while the other L2(11) acts on only 11 of them. This is analogous to the previous cases. However, the L2(11) generated by =, fl, 7 is maximal in M,2, while the L2(11) generated by =, fl, A is a subgroup of MlI, so there can be no question of an automorphism of M12 mapping these two groups onto each another. On the other hand we have an automorphism 0 defined by ao=ct-x, flo=fl, ~0=y, A0=A. (1.17) The 24 points of the projective line PL(23) can be labelled by ~ = {0 1 2 3... 22 ~} consisting of the marks of GF(23) and the symbol ~. The homography group L2(23 ) is generated by a:i~i+l, fl:i~2i, T:i~-l/i (1.18) (modulo 23). As permutations, =(0123456789 1011 1213141516 1718 192021 22), fl=(1248 16918 133612)(5 102017112221 19 15714), = (0 o0)(1 22)(211)(417)(820)(1610)(9 5)(1814)(13 7)(315)(619)(12 21) (1.19) Mathieu groups and Golay codes 157 The group is extended to the Mathieu group M24 by the supplementary generator 6 = (752120 10)(22 14171119)(13 9 12 816)(118 4 26). (1.20) In fact M24 is generated by ~, fl, and 76, since (76) 5 = ~ and (76)6 = 6. We have 76=(oo0)(315)(1144116221817219)(5122016792181013). (1.21) The generators ~t, fl, and 6 generate M23 (the one-point stabilizer fixing the symbol ~). M24 is the automorphism group of the Steiner system S(5, 8, 24). The automorphism group is transitive on the 759 special octads of the Steiner system, and on the 2576 umbral dodecads (an umbral dodecad is constructed as the symmetric difference of two special octads with just two common symbols).
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