Local Subgroups of the Monster and Odd Code Loops

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)).