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Local Subgroups of the Monster and Odd Code Loops

1995; American Mathematical Society; Volume: 347; Issue: 5 Linguagem: Inglês

10.2307/2154959

1088-6850

Thomas M. Richardson,

Genomic variations and chromosomal abnormalities

The main result of this work is an explicit construction of p-local subgroups of the Monster, the largest sporadic simple group.The groups constructed are the normalizers in the Monster of certain subgroups of order 32 , 52 , and 72 and have shapes 32+5+10-(Af11 xGL(2, 3)), 52+2+4-(S3xGL(2, 5)), and 72+1+2 • GL(2, 7).These groups result from a general construction which proceeds in three steps.We start with a self-orthogonal code C of length n over the field Fp , where p is an odd prime.The first step is to define a code loop L whose structure is based on C .The second step is to define a group N of permutations of functions from F2 to L. The final step is to show that N has a normal subgroup K of order p2 .The result of this construction is the quotient group N/K of shape p2+m+2m(S x GL(2,p)), where m + 1 = dim(C) and S is the group of permutations of Aut(C).To show that the groups we construct are contained in the Monster, we make use of certain lattices A(C), defined in terms of the code C .One step in demonstrating this is to show that the centralizer of an element of order p in N/K is contained in the centralizer of an element of order p in the Monster.The lattices are useful in this regard since a quotient of the automorphism group of the lattice is a composition factor of the appropriate centralizer in the Monster.This work was inspired by a similar construction using code loops based on binary codes that John Conway used to construct a subgroup of the Monster of shape 22+11+22 • (M24 x GL(2, 2)).