On the classification of extremely primitive affine groups

TL;DR

This paper completes the classification of extremely primitive affine groups without relying on Wall's conjecture, using recent advances on regular orbits of almost simple groups acting on modules.

Contribution

It introduces a new approach that fully classifies extremely primitive affine groups independently of Wall's conjecture, building on recent work on regular orbits.

Findings

Complete classification of extremely primitive affine groups achieved.

New method avoids dependence on Wall's conjecture.

Utilizes recent results on regular orbits of almost simple groups.

Abstract

Let $G$ be a finite non-regular primitive permutation group on a set $\Omega$ with point stabiliser $G_{\alpha}$. Then $G$ is said to be extremely primitive if $G_{\alpha}$ acts primitively on each of its orbits in $\Omega \setminus \{\alpha\}$, which is a notion dating back to work of Manning in the 1920s. By a theorem of Mann, Praeger and Seress, it is known that every extremely primitive group is either almost simple or affine, and all the almost simple examples have subsequently been determined. Similarly, Mann et al. have classified all of the affine extremely primitive groups up to a finite, but undetermined, collection of groups. Moreover, if one assumes Wall's conjecture on the number of maximal subgroups of an almost simple group, then there is an explicit list of candidates, each of which has been eliminated in a recent paper by Burness and Thomas. So, modulo Wall's conjecture, the classification of extremely primitive groups is complete. In this paper we adopt a different approach, which allows us to complete this classification in full generality, independent of the veracity or otherwise of Wall's conjecture in the almost simple setting. Our method relies on recent work of Fawcett, Lee and others on the existence of regular orbits of almost simple groups acting on irreducible modules.