A New Maximal Subgroup of $E_8$ in Characteristic $3$

TL;DR

This paper establishes the existence and uniqueness of a previously unrecognized maximal subgroup of the algebraic group $E_8$ in characteristic 3, specifically of type $F_4$, and explores its embedding properties.

Contribution

It identifies a new maximal subgroup of $E_8$ of type $F_4$ in characteristic 3, filling a gap in the classification of maximal subgroups.

Findings

Existence and uniqueness of the new $F_4$-type subgroup in $E_8$ in characteristic 3

Characterization of embeddings of $^3D_4(2)$ into $E_8$ with specific module composition factors

The new subgroup was missing from previous classifications by Seitz and Liebeck--Seitz

Abstract

We prove the existence and uniqueness of a new maximal subgroup of the algebraic group of type $E_8$ in characteristic $3$. This has type $F_4$, and was missing from previous lists of maximal subgroups produced by Seitz and Liebeck--Seitz. We also prove a result about the finite group $H={}^3\!D_4(2)$, that if $H$ embeds in $E_8$ (in any characteristic $p$) and has two composition factors on the adjoint module then $p=3$ and $H$ lies in this new maximal $F_4$ subgroup.