Odd Characterizations of Almost Simple Groups (PhD Thesis 2009)

TL;DR

This thesis develops methods for recognizing finite groups through local subgroup structures, character theory, and block theory, successfully characterizing several almost simple groups including b9(9), b7(8)(2), and b7(23).

Contribution

It introduces new character and local subgroup techniques for recognizing almost simple groups, with specific characterizations of b9(9), b7(8)(2), and b7(23).

Findings

Characterization of b9(9) using normalizers of 3-subgroups.

Description of local structure of groups of characteristic 3.

3-local characterization of the sporadic group b7(23).

Abstract

In this PhD thesis we discuss methods of recognizing finite groups by the structure of normalizers of certain 3-subgroups. We explain a method for characterizing groups using character theoretic and block theoretic methods and we use these methods to characterize $\alt(9)$. Furthermore, we describe a particular hypothesis related to the 3-local structure of finite groups of local characteristic 3 and characterize two almost simple proper extensions of $\Omega^+_8(2)$ as examples of the local approach to group recognition. We also give a 3-local characterization of the sporadic simple group $\HN$ using local methods whilst applying a character theoretic result.