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

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