# Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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## Local subgroups of the Monster and odd code loopsHTML articles powered by AMS MathViewer

by Thomas M. Richardson
Trans. Amer. Math. Soc. 347 (1995), 1453-1531 Request permission

## 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 ${2^{2 + 11 + 22}}\cdot ({M_{24}} \times GL(2,2))$.
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