Aschbacher's Theorem for the General Linear Group

TL;DR

This paper provides an accessible, comprehensive proof of Aschbacher's classification of maximal subgroups of the general linear group, detailing nine classes with clear descriptions and the properties that define them.

Contribution

It offers a detailed, understandable exposition of Aschbacher's theorem specifically for the general linear group, including descriptions of all nine subgroup classes and their properties.

Findings

Classifies maximal subgroups into nine classes

Describes eight classes based on stabilizing objects

Identifies properties of the ninth class with a unique normal quasisimple subgroup

Abstract

In 1984, Michael Aschbacher proved a seminal classification theorem for the maximal subgroups of effectively all of the classical groups. In this thesis we give a comprehensive, yet accessible description and proof of Aschbacher's theorem, restricting its scope to the general linear group. The main theorem of this paper classifies the maximal subgroups of the general linear group into nine different classes; eight of which have natural descriptions based on an object that their members act on and stabilise, whilst the ninth class - though not having such a natural description - contains groups that are bound by the property of having a unique normal quasisimple subgroup that acts absolutely irreducibly on the vector space. We give a detailed description of each of the first eight classes before proving that if a subgroup is not contained in a member of one of them, then it must have the properties that make up the ninth class. This paper uses techniques that cross over the fields of group theory, linear algebra and representation theory and it is approachable for anyone with an undergraduate understanding of these subjects.