Local subgroups of the Monster and odd code loops.

Abstract

The main result of this thesis 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 $3\sp2,5\sp2,$ and 7$\sp2$, and have shapes $3\sp{2 + 5 + 10}.(M\sb{11} \times GL(2,3)),\quad 5\sp{2 + 2 + 4}.(S\sb3 \times GL(2,5)),\enspace {\rm and}\quad 7\sp{2 + 1 + 2}.GL(2,7).$ 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 ${\rm I\!F}\sb{p}$, 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 ${\rm I\!F}\sbsp{p}{2}$ to L. The final step is to show that N has a normal subgroup K of order $p\sp2$. The result of this construction is the quotient group $N/K$ of shape $p\sp{2 + m + 2m}(S \times GL(2,p))$, where $m + 1 = dim(C)$ and S is the group of permutations of $Aut(C).$. There is an interesting relationship between $N/K$ and a lattice. Let $\varepsilon$ be a primitive $p\sp{\rm th}$ root of 1. There is a lattice $\Lambda$ consisting of the points in $\doubz\lbrack \varepsilon\rbrack \sp{n}$ which satisfy some congruences expressed in terms of the code C. Now $N/K$ has a subgroup M such that M has a normal subgroup Q which is an extraspecial p-group, $M/Q$ is a homomorphic image of the group of monomial automorphisms of $\Lambda$, and $Q/Z(Q)$ is isomorphic to $\Lambda/(\varepsilon - 1)\Lambda$ as an $M/Q$ module. A key step in showing that the three groups listed above are subgroups of the Monster is showing that M is isomorphic to a subgroup of the centralizer of an element of order p in the Monster, which we call A. In each case, A has a normal subgroup isomorphic to Q, and $A/Q$ is a homomorphic image of the group of all automorphisms of $\Lambda$. 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 $2\sp{2 + 11 + 22}.(M\sb{24} \times GL(2,2)).$.