On derangements in simple permutation groups

TL;DR

This paper investigates derangements in finite simple and primitive permutation groups, establishing new lower bounds on their proportion and demonstrating their generative properties, thus extending and refining previous results in the field.

Contribution

It provides improved lower bounds on derangement proportions and shows that simple transitive groups can be generated by two conjugate derangements, advancing understanding of derangements in permutation groups.

Findings

Proves $ ext{delta}(G) extgreater 89/325$ for certain simple groups.

Shows $G = ext{Delta}(G)^2$ for all large simple transitive groups.

Demonstrates that every finite simple transitive group can be generated by two conjugate derangements.

Abstract

Let $G \leqslant {\rm Sym}(\Omega)$ be a finite transitive permutation group and recall that an element in $G$ is a derangement if it has no fixed points on $\Omega$. Let $\Delta(G)$ be the set of derangements in $G$ and define $\delta(G) = |\Delta(G)|/|G|$ and $\Delta(G)^2 = \{ xy \,:\, x,y \in \Delta(G)\}$. In recent years, there has been a focus on studying derangements in simple groups, leading to several remarkable results. For example, by combining a theorem of Fulman and Guralnick with recent work by Larsen, Shalev and Tiep, it follows that $\delta(G) \geqslant 0.016$ and $G = \Delta(G)^2$ for all sufficiently large simple transitive groups $G$. In this paper, we extend these results in several directions. For example, we prove that $\delta(G) \geqslant 89/325$ and $G = \Delta(G)^2$ for all finite simple primitive groups with soluble point stabilisers, without any order assumptions, and we show that the given lower bound on $\delta(G)$ is best possible. We also prove that every finite simple transitive group can be generated by two conjugate derangements, and we present several new results on derangements in arbitrary primitive permutation groups.