LOCAL SUBGROUPS OF THE MONSTER AND ODD CODE LOOPS

The main result of this work is an explicit construction of p-local su bgroups of the Monster, the largest sporadic simple group. The groups constructed are the normalizers in the Monster of certain subgroups of order 3(2), 5(2), and 7(2) and have shapes 3(2+5+10).(M(11) x GL(2,3) ), 5(2+2+4).(S-3 x GL(2, 5)), and 7(2+1+2). GL(2, 7). These groups res ult from a general construction which proceeds in three steps. We star t with a self-orthogonal code C of length n over the field F-p, where p is an odd prime. The first step is to define a code loop L whose str ucture is based on C. The second step is to define a group N of permut ations of functions from F-p(2) to L. The final step is to show that N has a normal subgroup K of order p(2). The result of this constructio n is the quotient group N/K of shape p(2+m+2m)(S x GL(2, p)), where m + 1 = dim(C) and S is the group of permutations of Aut(C). To show tha t the groups we construct are contained in the Monster, we make use of certain lattices Lambda(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 quo tient 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 2(211+22).(M(24) x GL(2, 2)).