A note on extremely primitive affine groups

TL;DR

This paper confirms that all affine groups are not extremely primitive, completing the classification of extremely primitive groups under certain conjectural assumptions, and thus advancing understanding of primitive permutation groups.

Contribution

The paper proves that no affine primitive groups are extremely primitive, confirming a conjecture and completing the classification of extremely primitive groups under Wall's conjecture.

Findings

Confirmed that affine groups are not extremely primitive

Completed the classification of extremely primitive groups

Supported the conjecture that affine candidates are not extremely primitive

Abstract

Let $G$ be a finite primitive permutation group on a set $\Omega$ with nontrivial point stabilizer $G_{\alpha}$. We say that $G$ is extremely primitive if $G_{\alpha}$ acts primitively on each of its orbits in $\Omega \setminus \{\alpha\}$. In earlier work, Mann, Praeger and Seress have proved that every extremely primitive group is either almost simple or of affine type and they have classified the affine groups up to the possibility of at most finitely many exceptions. More recently, the almost simple extremely primitive groups have been completely determined. If one assumes Wall's conjecture on the number of maximal subgroups of almost simple groups, then the results of Mann et al. show that it just remains to eliminate an explicit list of affine groups in order to complete the classification of the extremely primitive groups. Mann et al. have conjectured that none of these affine candidates are extremely primitive and our main result confirms this conjecture.