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Constructing a maximal cofinitary group. (English) Zbl 1028.20001
A subgroup $$G$$ of the symmetric group $$\text{Sym}(\omega)$$ is ‘cofinitary’ if every non-identity element of $$G$$ has just finitely many fixed points. See [P. J. Cameron, Bull. Lond. Math. Soc. 28, No. 2, 113-140 (1996; Zbl 0853.20002)] for a survey of cofinitary groups. By Zorn’s Lemma, every cofinitary group is contained in a ‘maximal cofinitary group’, that is, one maximal among the cofinitary subgroups of $$\text{Sym}(\omega)$$. Assuming CH, the author gives a rather more concrete step-by-step construction of a maximal cofinitary group. He gives an analogous construction under Martin’s Axiom and the negation of CH.

##### MSC:
 20B07 General theory for infinite permutation groups 20E28 Maximal subgroups 20B35 Subgroups of symmetric groups 03E50 Continuum hypothesis and Martin’s axiom
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