Permutation groups of prime power degree and $p$-complements

TL;DR

This paper classifies almost simple groups with prime power degree permutation representations and explores properties of $p$-complements, revealing structural insights and extending known examples of inequivalent complements.

Contribution

It provides a classification of almost simple groups with prime power degree actions and analyzes the automorphism inequivalence of $p$-complements, extending previous examples.

Findings

Primitive permutation groups of prime power degree have a regular subgroup.

Any two faithful primitive representations of the same prime power degree are automorphically equivalent.

The number of inequivalent $p$-complements can be arbitrarily large.

Abstract

Extending earlier work of Guralnick and of Cai and Zhang, we classify the almost simple groups which have transitive permutation representations of prime power degree $p^k$, and those which have $p$-complements (stabilisers of order coprime to $p$ in such representations). We deduce that every primitive permutation group of prime power degree has a regular subgroup, and that any two faithful primitive representations of a group, of the same prime power degree, are equivalent under automorphisms. In general, $p$-complements in a finite group can be inequivalent under automorphisms, or even non-isomorphic. We extend examples of such phenomena due to Buturlakin, Revin and Nesterov by showing that the number of inequivalent classes of complements can be arbitrarily large. Questions concerning the existence of prime power representations and $p$-complements in groups with socle ${\rm PSL}_d(q)$ are related to some difficult open problems in Number Theory.