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A Representation of the Mathieu Group <Emphasis Type= A representation of the Mathieu group ;1I~ as a, collineation group. by J-. A. TODD (Cambridge, England) In memory of Guido Castelnuovo, in the recurrence of the first centenary of his birth. Summary. - The Mathieu group M~4 can be represented as a collineation group in space of 11 dimensions over the field of t~vo elements. This paloer discusses the geometry associated .with this representation. Intl'oduetion. The quintuply transitive MATHI]~U group M24, of degree 24 and order 24.23.22.21.20.48--~ 244, 823, 040, is the automorphism group of a ST]~INER system S(5, 8, 24), that is, an arrangement of 24 objects in sets of 8, such that any 5 objects belong to exactly one set. It has been shown by WI~'T [5] that such a system is essentially unique. Some years ago I proved [4] that M2~ could be represented as a eollineation group in a projective space of 11 dimensions over the field GF(2). The object of this paper is to develop the properties of this representation in some detail, and in particular to show how a number of maximal subgroups of M2~ can be interpreted geometrically in a very simple manner. In § 1 below we describe the main properties of the STEn~ER system S(5, 8, 24) which will be used in the sequel. The projective space is introduced in §2. In §3 we describe the individual operations of the group, and their invariant subspaees when the group is represented as a collineation group. Finally, in § 4, we describe 8 maximal subgroups of M~. In the ten tables which follow the paper we give, first, an explicit form of the S~:EIN]~R system S(5, 8, 24) and then the character tables of M24 and the 8 maximal subgroups. § 1. - The Steiner system S(5, 8, 24). 1.1. - The S~E]z~]~t~ system S(5, 8, 24) is an arrangement of 24 elements in sets of 8, such that any 5 of the elements belong to exactly one set. We assume WI~T's result [5] that such a system is unique to within isomorphism. 200 J. A. TODD: A representatio~ of the Mathieu group M:~, etc. We shall refer to the sets of S elements under discussion as octads. The number of octads in the system is (~/1~}~4 s or 759. Each of the 24 elements belongs to 759.8/24 or 253 octads, each pair of elements to 253.7/23 or 77, each triad to 77.6/22 or 21 and each tetrad to~ 21.5/21 or 5. If ~ is a given octad, there are 4 further octads containing each tetrad in ~, so that 280 octads share a tetrad with ~. A simple computation shows that any octad containing 3 elements of a must be among these 280, and hence contain a fourth member of a. Of the 76 octads, other than ~, which contain an assigned pair of elements of ~, 60 belong to this set of 280 (4 through each of the 15 pairs formed from the other 6 elements of ~). Hence there remain 16 octads containing a given pair of elements of ~ and no further elements of ~, and hence 16.28 or 448 octads having some pair of elements in common with ~. It is easy to verify that every octad containing an element of a must contain at least one further element of a. Thus there remain 30 octads~ the elements in any one of which are distinct from those of ~. A method of constructing a STEINER system S(5, 8, 24) is given by CAR~C~AEL [2, 432]. The 24 elements are represented by the marks 0, 1, ..., 22 of the Galois field GFt23), together with the symbol ~. The linear fractional group LF(2, 23) acts on these symbols, and the set of elements (~0 1 3 12 15 21 22) is left unaltered by a subgroup of order 8 generated by x'-~(l+x)/(1--x)and • x'-----(3x +ll/(x--3). The 759 octads are obtained from the set (~ 01 3 12 15 21 22) by applying the transformations of LF(2, 23). These octads are set out at length in Table I at the end of this paper, and we shall have frequent occa- sion to refer to this table. The group M~, which is the automorphism group of the system, permutes the octads transitively. 1.2.- We have seen that any pair of octads have 0, 2 or 4 common elements. Two tetrads of elements which together constitute an octad will be called complementary. Since each tetrad belongs to five octads, it deter- miaes five complementary tetrads. A basic property of S(5, 8, 24) is that any two of these tetrads are themselves complementary. For suppose that -~ (ala2asa~asa~aTas) and ~ ~ {ala~a3a4agaloal~al~) are two octads containing ala2aaa4. Suppose that aso6aTasaga~oa~a~2 do not form an octad, we shall derive a contradiction. There is a unique octad, ~, say, containing asasaTasag. This octad must have an even number of elements in common with 9, and it cannot contain any of a~a2a~a~ since it would then have five common elements with a. Hence, under our hypothesis that a~a6a~a6aga~oa~a~2 is not an oetad, 7 must contain one of the elements a~oa~a~2, say a~o. In a similar way, the octad ~, containing a~a6a~asa~ must contain one of the elements aoa~oa~2, and this can only be a~, otherwise y and ~ have five common elements. Consider, now, the octad ~ containing asasa~a~a~. This d. A. TODD: A_ representatio~ of the Mathieu group M24, etc. 201 must have four members in common with a, and the fourth one conner be as, otherwise T and ¢ have five common members. Hence ~ contains one (and only one) of a~a2asa,, say al. Then ~ and ~, having in common ala,a~ must have a further common element; this must be a.~o or a~ and then ~ has five common members with y or ~. This contradiction shows that a~a6aTasaoa~sa~a~ must be an octad. Thus the five tetrads complementary to a given tetrad are themselves complementary in pairs, and the (~) tetrads formed from the 24 elements fall, in a unique way, into sets of six mutually complementary tetrads. There are 1771 such sets, permuted transitively by the operations of M24. From Table I we see that the tetrads c~0 1 2, 3 5 14: 17, 4 13 16 22, .6 7 19 21, 8 11 12 18, 9 10 15 20 form a set of the type in question. If ~ and ~ are two octads with no common elements, the eight elements not included in a and ~ themselves form an octad. For suppose the contrary, and denote the elements by ala2asa4asasa~as. The octad containing any five of these elements, since it has an even number of elements in common with :¢ and ~, must contain a sixth element of the set. Suppose that the octad y containing a~a2a~a4a5 contains as, then the octad ~ containing ala2a3a4aT, since it eanna t have more than four elements in common with T, must contain as. The octad e containing ala2aaasa7 must then contain a4, a6 or as and thus have five elements in common with either T or ~. This contradiction shows that ala2aaa~asct6a~a8 constitute an octad. A set of three mutually disjoint octads will be calted a trio. The 30 octads disjoint from a given octad fall into 15 disjoint pairs, so that each oetad belongs to 15 trios. There are thus 3795 trios, and they are permuted transitively by the operations of M24. The six tetrads in a mutually complementary set can be combined in three pairs to form a trio of octads in 15 different ways. Of the 15 trios containing a given octad a, three arise in this way from each of the 35 sepa- rations of the elements of a into two tetrads. It follows that, if a, ~, y are three octads forming a trio, there are determined, in each of the oetads, 7 pairs of complementary tetrads, and that these belong to 7 sets of mutually complementary tetrads which combine in pairs to give the octads a, ~, ? of the trio. Thus, for the trio of oetads c~0 1 2 3 5 14 17, 4 6 7 13 16 19 21 22, 8 9 10 11 12 15 18 20 we have the following seven sets of mutually complementary tetrads which combine in pairs to form the three octads. Annali gi Matemativa 26 202 J. A. TODD: A representatio~ o] the Mathieu group Mu, etc. c<)0 1 2 3 5 14 17 4 13 16 22 6 7 19 21 8 11 12 18 9 10 15 20 c~0 3 14 I 2 5 17 4 6 7 22 13 16 19 21 8 15 18 20 9 10 11 12 c~0 5 17 1 2 3 14 4 19 21 22 6 7 13 16 8 9 10 18 11 12 15 20 c<)l 3 17 0 2 5 14 4 7 13 21 6 16 19 22 8 9 11 15 10 12 18 20 c<>l 5 14 0 2 3 17 4 6 13 19 7 16 21 22 8 10 11 20 9 12 15 18 ¢:~2 3 5 0 1 14 17 4 7 16 19 6 13 21 22 8 10 12 15 9 11 18 20 cx~2 14 17 0 1 3 5 4 6 16 21 7 13 19 22 8 9 12 20 10 11 15 18 It will be observed that the 7 pairs of complementary tetrads belonging to each octad are disposed in such a a manner that any two tetrads belonging to different pairs have two common elements.
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