On Non-Generic Finite Subgroups of Exceptional Algebraic Groups

TL;DR

This paper classifies non-generic finite subgroups of exceptional algebraic groups, showing many, including large symmetric and alternating groups, cannot be Lie primitive or maximal subgroups, thus clarifying their subgroup structure.

Contribution

It provides a classification of non-generic subgroups in exceptional algebraic groups, identifying which almost simple groups can or cannot occur as Lie primitive or maximal subgroups.

Findings

Large symmetric and alternating groups for n ≥ 10 do not occur as Lie primitive subgroups.

A short list of possible non-G-completely reducible, non-generic simple subgroups is derived.

Feasible characters restrict possible composition factors of subgroups in exceptional groups.

Abstract

The study of finite subgroups of a simple algebraic group $G$ reduces in a sense to those which are almost simple. If an almost simple subgroup of $G$ has a socle which is not isomorphic to a group of Lie type in the underlying characteristic of $G$, then the subgroup is called non-generic. This paper considers non-generic subgroups of simple algebraic groups of exceptional type in arbitrary characteristic. A finite subgroup is called Lie primitive if it lies in no proper subgroup of positive dimension. We prove here that many non-generic subgroup types, including the alternating and symmetric groups $\text{Alt}_{n}$, $\text{Sym}_{n}$ for $n \ge 10$, do not occur as Lie primitive subgroups of an exceptional algebraic group. A subgroup of $G$ is called $G$-completely reducible if, whenever it lies in a parabolic subgroup of $G$, it lies in a conjugate of the corresponding Levi factor. Here, we derive a fairly short list of possible isomorphism types of non-$G$-completely reducible, non-generic simple subgroups. As an intermediate result, for each simply connected $G$ of exceptional type, and each non-generic finite simple group $H$ which embeds into $G/Z(G)$, we derive a set of feasible characters, which restrict the possible composition factors of $V \downarrow S$, whenever $S$ is a subgroup of $G$ with image $H$ in $G/Z(G)$, and $V$ is either the Lie algebra of $G$ or a non-trivial Weyl module for $G$ of least dimension. This has implications for the subgroup structure of the finite groups of exceptional Lie type. For instance, we show that for $n \ge 10$, $\text{Alt}_n$ and $\text{Sym}_n$, as well as numerous other almost simple groups, cannot occur as a maximal subgroup of an almost simple group whose socle is a finite simple group of exceptional Lie type.