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