On Certain Small Lie Rank Subgroups of $E_8(2)$

Abstract

This thesis forms a part of the much lager project whose aim is to classify the maximal subgroups of the finite simple exceptional group of Lie type $E_8(2)$. Groups $H$ with $F^*(H)$ isomorphic to $L_2(64)$, $L_2(16)$, $L_2(8)$, $L_3(4)$, or $L_3(3)$ arise as some of the possible candidates for maximal subgroups of $E_8(2)$. We prove that if $F^*(H)$ is isomorphic to $L_2(64)$, $L_2(16)$ or $L_3(4)$ then $H$ cannot be maximal in $E_8(2)$. Partial progress is made towards establishing whether $L_2(8)$ can be a maximal subgroup. A highlight is that we find maximal subgroups of $E_8(2)$ isomorphic to $L_3(3)$; we show that there are at most $3$ conjugacy classes of them. Extensive use of the computer algebra package \textsc{Magma} has been made to prove our results. After the work done in this thesis not much is left to do in order to classify the maximal subgroups of $E_8(2)$.

Date of Award	1 Aug 2022
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Peter Rowley (Supervisor) & Yuri Bazlov (Supervisor)