Cofinitary Permutation Groups

Abstract

A permutation group is cofinitary if any non-identity element fixes only finitely many points. This paper presents a survey of such groups. The paper has four parts. Sections 1–6 develop some basic theory, concerning groups with finite orbits, topology, maximality, and normal subgroups. Sections 7–12 give a variety of constructions, both direct and from geometry, combinatorial group theory, trees, and homogeneous relational structures. Sections 13–15 present some generalisations of sharply k-transitive groups, including an orbit-counting result with a character-theoretic flavour. The final section treats some miscellaneous topics. Several open problems are mentioned.