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Isomorphism Types of Maximal Cofinitary Groups

Published online by Cambridge University Press:  15 January 2014

Bart Kastermans
Bart Kastermans*
Affiliation:
Department of Mathematics, University of Wisconsin, Madison, WI 53706, USAE-mail: kasterma@math.wisc.edu
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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.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 15 , Issue 3 , September 2009 , pp. 300 - 319
Copyright
Copyright © Association for Symbolic Logic 2009

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References

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