Prime degree irreducible representations of simple algebraic groups and finite simple groups of Lie type

Abstract

Let $r,p$ be primes and $k$ be a positive integer. We show that if $G$ is a subgroup of $S{L}_r(p^k)$ lying only in the non-geometric Aschbacher class ${\mathcal{C}}_9$, then $|G|$ is bounded above by a function of $r$ and $k$, independently of $p$. We also show that, up to conjugacy in $G{L}_r(p^k)$, the number of such $G$ is bounded above by a function of $r$ that is independent of $p$ and $k$. Apart from being of interest in their own right, these results have an application in a computational version of the strong approximation theorem for finitely generated Zariski-dense subgroups of $S{L}_r(\mathbb{P})$, where $\mathbb{P}$ is a number field.