Fixers and derangements of finite permutation groups

TL;DR

This paper classifies large fixers in certain finite permutation groups, explores their properties, and verifies a related conjecture, advancing understanding of group actions and fixed point structures.

Contribution

It provides a classification of large fixers in almost simple primitive groups with socle PSL2(q) and verifies a case of Spiga's conjecture on permutation characters.

Findings

Classified large fixers in groups with socle PSL2(q).

Verified a case of Spiga's conjecture.

Presented results on fixers in groups with alternating or sporadic socles.

Abstract

Let $G\leqslant\mathrm{Sym}(\Omega)$ 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 $|K| \geqslant |H|$. There is a special interest in studying large fixers due to connections with Erd\H{o}s-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.