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Cofinitary Permutation Groups
Author(s) -
Cameron Peter J.
Publication year - 1996
Publication title -
bulletin of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 2.396
H-Index - 48
eISSN - 1469-2120
pISSN - 0024-6093
DOI - 10.1112/blms/28.2.113
Subject(s) - mathematics , permutation group , group (periodic table) , transitive relation , permutation (music) , section (typography) , homogeneous , orbit (dynamics) , variety (cybernetics) , pure mathematics , group action , combinatorics , discrete mathematics , computer science , chemistry , physics , statistics , organic chemistry , acoustics , engineering , aerospace engineering , operating system
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.