Derangements in primitive permutation groups, with an application to character theory

Abstract

Let $G$ be a finite primitive permutation group and let $\kappa (G)$ be the number of conjugacy classes of derangements in $G$. By a classical theorem of Jordan, $\kappa (G)⩾1$. In this paper we classify the groups $G$ with $\kappa (G)=1$, and we use this to obtain new results on the structure of finite groups with an irreducible complex character that vanishes on a unique conjugacy class. We also obtain detailed structural information on the groups with $\kappa (G)=2$, including a complete classification for almost simple groups.

29 pages