Alternating Subgroups of Exceptional Groups of Lie Type

TL;DR

This paper investigates the embeddings of alternating and symmetric groups into almost simple groups of exceptional Lie type, establishing non-existence of certain maximal subgroups except in specific small cases.

Contribution

It proves that, except for degrees 6 and 7, no maximal subgroup of an almost simple exceptional Lie type group is an alternating or symmetric group, advancing understanding of subgroup structure.

Findings

No maximal alternating or symmetric subgroups for degrees other than 6 or 7.

Provides detailed information on embeddings in the remaining open cases.

First comprehensive analysis of these subgroup embeddings in exceptional groups.

Abstract

In this paper we examine embeddings of alternating groups and symmetric groups into almost simple groups of exceptional type. In particular, we prove that unless the alternating or symmetric group has degree 6 or 7, there is no maximal subgroup of any almost simple group with socle an exceptional group of Lie type that is an alternating or symmetric group. Furthermore, in the remaining open cases we give considerable information about the possible embeddings. Note that no maximal alternating or symmetric subgroups are known in the remaining cases. This is the first in a sequence of papers aiming to substantially improve the state of knowledge about the maximal subgroups of exceptional groups of Lie type.