Isomorphism types of maximal cofinitary groups

Bulletin of Symbolic Logic 15 (3):300-319 (2009)
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Abstract

A cofinitary group is a subgroup of Sym(ℕ) where all nonidentity elements have finitely many fixed points. A maximal cofinitary group is a cofinitary group, maximal with respect to inclusion. We show that a maximal cofinitary group cannot have infinitely many orbits. We also show, using Martin's Axiom, that no further restrictions on the number of orbits can be obtained. We show that Martin's Axiom implies there exist locally finite maximal cofinitary groups. Finally we show that there exists a uniformly computable sequence of permutations generating a cofinitary group whose isomorphism type is not computable

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References found in this work

Happy Families.A. R. D. Mathias - 1977 - Annals of Mathematical Logic 12 (1):59.
Infinite Combinatorics and Definability.Arnold W. Miller - 1989 - Annals of Pure and Applied Logic 41 (2):179-203.
Maximal Cofinitary Groups.Yi Zhang - 2000 - Archive for Mathematical Logic 39 (1):41-52.
Cardinal Invariants Related to Permutation Groups.Bart Kastermans & Yi Zhang - 2006 - Annals of Pure and Applied Logic 143 (1-3):139-146.

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