Abstract: Let G be a finite transitive permutation group on Ω, is a fixer if every element of fixes some point in Ω. We call a fixer K large if |K|≥|G_ω|, and maximal if any proper overgroup of K is not a fixer. The study of fixers of permutation groups is related to the Erdös-Ko-Rado problem for permutation groups, derangements of permutation groups, and conjugacy classes of finite groups. We characterize large maximal fixers of primitive two-dimensional linear groups and primitive acions of simple group of twisted Lie rank 1. We will also give some examples of fixers of almost simple groups.
