Fixers and derangements of finite permutation groups

Abstract

Let $G⩽\mathrm{Sym}(\Omega )$ 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 Erdős-Ko-Rado type problems. In this paper, we classify up to conjugacy the large fixers of the almost simple primitive groups with socle ${\mathrm{PSL}}_2(q)$, 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.

33 pages, to appear in J. Algebraic Combin