On the Maximal Subgroups of $\mathrm{E}_8(3)$ and $\mathrm{E}_8(2)$

Abstract

This thesis forms part of a larger project which aims to complete the classification of the maximal subgroups of the finite exceptional simple groups of Lie type $\E{8}{3}$ and $\E{8}{2}$. A classification theorem for the maximal subgroups of the finite exceptional simple groups of Lie type can be found in Liebeck and Seitz \cite{liebeckseitzsurvey}. Using this theorem, one can reduce the problem of classifying the maximal subgroups of such groups to studying the maximality of a list of almost simple groups. Tables \ref{Lie3grps} and \ref{notLie3grps} list these possibilities for $\E{8}{3}$. In this thesis, we predominantly deal with the cases $\PSL{2}{3^n}$ for $n \in \{3,4,5,6,7\}$, $\PSL{3}{5}$, $\PSL{2}{11}$ and $\mathieu{11}$ inside $\E{8}{3}$. For the groups $H\cong\PSL{2}{81}$ and $H\cong\PSL{3}{5}$, we construct representatives for all conjugacy classes of subgroups inside $\E{8}{3}$ isomorphic to $H$. For all groups mentioned here, we find that none of them exhibit maximal embeddings into $\E{8}{3}$ apart from $\PSL{3}{5}$ (and $\PSL{2}{27}$, where this work is unfinished). In the case of $H\cong\PSL{3}{5}$, we find that $\E{8}{3}$ contains one conjugacy class of subgroups isomorphic to $H$ and that these subgroups are maximal. Theorem \ref{summarythm} concisely summarises the current state of affairs for the classification of the maximal subgroups of $\E{8}{3}$. In addition to this work, we also consider the case $\PSL{2}{8} < \E{8}{2}$; this is a continuation of the work of Javed \cite{mahah} and Neuhaus \cite{neuhaus}. Brauer characters and modular representation theory play a key role in this research. As such, Brauer character tables (over characteristic 3) for all groups mentioned here and more are given. Furthermore, a comprehensive report on the maximal tori of the finite exceptional simple groups of Lie type is given; this aligns with the work shown in my publication with Javed, Parkin and Rowley \cite{maxtori}. In this thesis, we give extra examples which are not present in the paper. The computer package \textsc{Magma} was used extensively throughout this research and the appendix contains all supporting code for this work.

Date of Award	14 Nov 2025
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Peter Rowley (Main Supervisor) & Richard Webb (Co Supervisor)