Primitive permutation groups and derangements of prime power order

TL;DR

This paper characterizes primitive permutation groups where all derangements have order a fixed prime power, revealing they are either almost simple or affine, with specific structural conditions.

Contribution

It classifies primitive groups with derangements of prime power order, identifying their structure as either almost simple or affine, and characterizes affine groups via two-point stabilizers.

Findings

Primitive groups with all derangements of prime power order are either almost simple or affine.

All almost simple groups with this property are classified.

Affine groups with this property have two-point stabilizers that are r-groups.

Abstract

Let $G$ be a transitive permutation group on a finite set of size at least $2$. By a well known theorem of Fein, Kantor and Schacher, $G$ contains a derangement of prime power order. In this paper, we study the finite primitive permutation groups with the extremal property that the order of every derangement is an $r$-power, for some fixed prime $r$. First we show that these groups are either almost simple or affine, and we determine all the almost simple groups with this property. We also prove that an affine group $G$ has this property if and only if every two-point stabilizer is an $r$-group. Here the structure of $G$ has been extensively studied in work of Guralnick and Wiegand on the multiplicative structure of Galois field extensions, and in later work of Fleischmann, Lempken and Tiep on $r'$-semiregular pairs.