Derangements in non-Frobenius groups

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 a conjecture of Guralnick--Tiep in large degree. We also give an application to coverings of varieties over finite fields.