Derangements in non-Frobenius groups

TL;DR

This paper establishes bounds on the proportion of derangements in large transitive permutation groups, showing that non-Frobenius groups have a significant derangement proportion, and confirms several conjectures in the field.

Contribution

It proves a sharp bound on derangements in large transitive groups, generalizing previous results and settling key conjectures in permutation group theory.

Findings

Derangement proportion exceeds 1/(2n^{1/2}) in non-Frobenius groups

Characterization of primitive Frobenius groups among large transitive groups

Application to finite field variety coverings

Abstract

We prove that if $G$ is a transitive permutation group of sufficiently large degree $n$, then either $G$ is primitive and Frobenius, or the proportion of derangements in $G$ is larger than $1/(2n^{1/2})$. This is sharp, generalizes substantially bounds of Cameron--Cohen and Guralnick--Wan, and settles conjectures of Guralnick--Tiep and Bailey--Cameron--Giudici--Royle in large degree. We also give an application to coverings of varieties over finite fields.