Derangements in Primitive Permutation Groups, With an Application to Character Theory

Let G be a finite primitive permutation group and let κ(G) be the number of conjugacy classes of derangements in G. By a classical theorem of Jordan, κ(G) > 1. In this paper we classify the groups G with κ(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 κ(G) = 2, including a complete classification for almost simple groups.