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Exotic Fusion Systems Related to Sporadic Simple Groups (Version With Code)

Martin van Beek

Abstract

We describe several exotic fusion systems related to the Sporadic simple groups at odd primes. More generally, we classify saturated fusion systems supported on Sylow $3$-subgroups of the Conway group $\mathrm{Co}_1$ and the Thompson group $\mathrm{F}_3$, and a Sylow $5$-subgroup of the Monster $\mathrm{M}$, as well as a particular maximal subgroup of the latter two $p$-groups. This work is supported by computations in MAGMA.

Exotic Fusion Systems Related to Sporadic Simple Groups (Version With Code)

Abstract

We describe several exotic fusion systems related to the Sporadic simple groups at odd primes. More generally, we classify saturated fusion systems supported on Sylow -subgroups of the Conway group and the Thompson group , and a Sylow -subgroup of the Monster , as well as a particular maximal subgroup of the latter two -groups. This work is supported by computations in MAGMA.
Paper Structure (7 sections, 91 theorems, 22 equations, 9 tables)

This paper contains 7 sections, 91 theorems, 22 equations, 9 tables.

Table of Contents

  1. Introduction
  2. Preliminaries: Groups
  3. Preliminaries: Fusion Systems
  4. Fusion Systems on a Sylow $3$-subgroup of $\mathrm{Co}_1$
  5. Fusion Systems related to a Sylow $3$-subgroup of $\mathrm{F}_3$
  6. Fusion Systems related to a Sylow $5$-subgroup of $\mathrm{M}$
  7. magma code for computations

Key Result

Proposition 2.2

[proposition]BasicJS Let $S$ be a non-trivial finite $p$-group. Then the following hold:

Theorems & Definitions (184)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • Proposition 2.4: Coprime Action
  • proof
  • Lemma 2.5: Burnside
  • Lemma 2.6
  • proof
  • Lemma 2.7: A$\times$B-Lemma
  • ...and 174 more