A new maximal subgroup of 𝐸₈ in characteristic 3

Craven, David; Stewart, David; Thomas, Adam

doi:10.1090/proc/15759

A new maximal subgroup of $E_8$ in characteristic $3$
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by David A. Craven, David I. Stewart and Adam R. Thomas PDF

Proc. Amer. Math. Soc. 150 (2022), 1435-1448 Request permission

Abstract:

We prove the existence and uniqueness up to conjugacy 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 $H={}^3\!D_4(2)$, namely 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 a conjugate of this new maximal $F_4$ subgroup.

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