Finite permutation groups with quasi-semiregular elements

TL;DR

This paper investigates the presence of quasi-semiregular elements in finite permutation groups, classifies primitive groups containing such elements, and provides a complete classification for certain almost simple groups.

Contribution

It classifies primitive permutation groups with quasi-semiregular elements and fully characterizes the almost simple groups with these elements when the socle is alternating or sporadic.

Findings

Primitive groups with quasi-semiregular elements are classified by O'Nan-Scott types.

Complete classification of almost simple groups with quasi-semiregular elements when socle is alternating or sporadic.

Reduces the problem to affine and almost simple cases for primitive groups.

Abstract

A quasi-semiregular element in a permutation group is an element that has a unique fixed point and acts semiregularly on the remaining points. Such elements were first studied in the context of automorphisms of graphs and occur naturally in many families of permutation groups, such as Frobenius and Zassenhaus groups. They also arise in the context of groups with a strongly $p$-embedded subgroup. We investigate the question of which finite permutation groups contain quasi-semiregular elements, with particular attention to the primitive permutation groups. We determine the O'Nan-Scott types of primitive groups that can contain quasi-semiregular elements and reduce the question to the affine and almost simple cases. In the almost simple case, we obtain a complete classification when the socle is alternating or sporadic.