On medium-rank Lie primitive and maximal subgroups of exceptional groups of Lie type

TL;DR

This paper investigates the existence and classification of certain primitive subgroups of exceptional algebraic groups of Lie type, showing that such subgroups are rare and mostly non-existent, with a few specific exceptions.

Contribution

It proves the non-existence of most Lie primitive subgroups in exceptional groups, except for a few known cases, and establishes their stability under automorphisms, advancing subgroup classification.

Findings

Most Lie primitive subgroups do not exist in exceptional groups

Identifies specific exceptions where primitive subgroups may occur

Shows that maximal subgroups correspond to fixed points of algebraic group automorphisms

Abstract

We study embeddings of groups of Lie type $H$ in characteristic $p$ into exceptional algebraic groups $\mathbf G$ of the same characteristic. We exclude the case where $H$ is of type $\mathrm{PSL}_2$. A subgroup of $\mathbf G$ is \emph{Lie primitive} if it is not contained in any proper, positive-dimensional subgroup of $\mathbf G$. With a few possible exceptions, we prove that there are no Lie primitive subgroups $H$ in $\mathbf G$, with the conditions on $H$ and $\mathbf G$ given above. The exceptions are for $H$ one of $\mathrm{PSL}_3(3)$, $\mathrm{PSU}_3(3)$, $\mathrm{PSL}_3(4)$, $\mathrm{PSU}_3(4)$, $\mathrm{PSU}_3(8)$, $\mathrm{PSU}_4(2)$, $\mathrm{PSp}_4(2)'$ and ${}^2\!B_2(8)$, and $\mathbf G$ of type $E_8$. No examples are known of such Lie primitive embeddings. We prove a slightly stronger result, including stability under automorphisms of $\mathbf G$. This has the consequence that, with the same exceptions, any almost simple group with socle $H$, that is maximal inside an almost simple exceptional group of Lie type $F_4$, $E_6$, ${}^2\!E_6$, $E_7$ and $E_8$, is the fixed points under the Frobenius map of a corresponding maximal closed subgroup inside the algebraic group. The proof uses a combination of representation-theoretic, algebraic group-theoretic, and computational means.