Fixers and derangements of finite permutation groups

Let G () be a finite transitive permutation group with point stabiliser H. We say that a subgroup K of G is a fixer if every element of K has fixed points, and we say that K is large if |K| |H|. There is a special interest in studying large fixers due to connections with Erdos-Ko-Rado type problems. In this paper, we classify up to conjugacy the large fixers of the almost simple primitive groups with socle PSL₂ (q), and we use this result to verify a special case of a conjecture of Spiga on permutation characters. We also present some results on large fixers of almost simple primitive groups with socle an alternating or sporadic group.