Local subgroups of the Monster and odd code loops

Abstract:

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 $3^{2}$, $5^{2}$, and $7^{2}$ and have shapes \[ {3^{2 + 5 + 10}}\cdot ({M_{11}} \times GL(2,3)),\quad {5^{2 + 2 + 4}}\cdot {S_3} \times GL(2,5)),\quad {\text {and}}{7^{2 + 1 + 2}}\cdot GL(2,7)\] . 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 ${\mathbb {F}_p}$, 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 $\mathbb {F}_p^2$ to $L$. The final step is to show that $N$ has a normal subgroup $K$ of order ${p^2}$. The result of this construction is the quotient group $N/K$ of shape ${p^{2 + m + 2m}}(S \times GL(2,p))$, where $m + 1 = \dim (C)$ and $S$ is the group of permutations of $\text {Aut}(C)$. To show that 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 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 ${2^{2 + 11 + 22}}\cdot ({M_{24}} \times GL(2,2))$.