By Murakami M.

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At the same time we establish many of the properties of these Steiner systems needed t o analyze the structure of the Mathieu groups and construct the Leech lattice, the Griess algebra, the Conway groups, and the Monster. The blocks in the Steiner system S(24,8,5) are called octads. In Section 19 we construct certain subgroups of the largest Mathieu group M24 and determine the action of these subgroups on the octads. Further we study two 11-dimensional GF(2)-modules for M24, the Golay code module and the Todd module.

D) for all d E L and all i = 1,2,3. Similarly as s is in the nucleus of L, K is also centralized by these elements. 4, KZ is centralized by $i(a) for each even a E I?. So K Z 5 Z(N+). Next let a E I' be odd. 5, [zi,$2(a)]= s and [$i(s),$2(a)l = $2(s)z3. Therefore Z a N , and N / C N ( Z ) S3. Therefore (3) holds and N+ = C N ( Z ) . Finally from the commutators in the previous paragraph, is centralized by +l(a)and [kl,&(a)] = k2, completing the proof of (2). E L}, and A = A 2 u A 3 . 4: (1) Q1 Nl = C N ( z l ) and kl € Z(N1).

J = A 5 N A U ~ ( X ) Let a1 : Y 4 Y be the identity map. Then we have achieved the hypothesis of paragraph one, so by that paragraph there exists an automorphism 7 of X with xy = y. Hence as NA2ct(Y)(C) = NAut(X)(Y) is t-transitive on Y , Aut(X) is (t 1)-transitive on X. Then as each (t 1)-subset of X is in a unique block, Aut(X) is also transitive on blocks. ~ then b = 1. So all orbits of B on C are regular. Thus NA(m) = E is a complement to B in A and C = mB. Further A, and hence also El is transitive on A and therefore also on m = { mnK : k f A}.

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