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

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 ${\mathrm{PSL}}_3(3)$, ${\mathrm{PSU}}_3(3)$, ${\mathrm{PSL}}_3(4)$, ${\mathrm{PSU}}_3(4)$, ${\mathrm{PSU}}_3(8)$, ${\mathrm{PSU}}_4(2)$, ${\mathrm{PSp}}_4(2{)}^{\prime }$ 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.

vi+214pp

Cited by in corpus (2)

• The maximal subgroups of the exceptional groups $F_4(q)$, $E_6(q)$ and ${}^2{E}_6(q)$ and related almost simple groups
• On fixed-point-free involutions in actions of finite exceptional groups of Lie type