Closed subgroups of the infinite symmetric group

by Bergman and Shelah. [BmSh:823]
Algebra Universalis, 2006
Let S= Sym (omega) be the group of all permutations of the natural numbers, and for subgroups G_1,G_2 <= S let us write G_1 approx G_2 if there exists a finite set U subseteq S such that < G_1 cup U>=< G_2 cup U> . It is shown that the subgroups closed in the function topology on S lie in precisely four equivalence classes under this relation. Given an arbitrary subgroup G <= S, which of these classes the closure of G belongs to depends on which of the following statements about pointwise stabilizer subgroups G_{(Gamma)} of finite subsets Gamma subseteq omega holds: (i) For every finite set Gamma, the subgroup G_{(Gamma)} has at least one infinite orbit in omega . (ii) There exist finite sets Gamma such that all orbits of G_{(Gamma)} are finite, but none such that the cardinalities of these orbits have a common finite bound. (iii) There exist finite sets Gamma such that the cardinalities of the orbits of G_{(Gamma)} have a common finite bound, but none such that G_{(Gamma)}= {1}. (iv) There exist finite sets Gamma such that G_{(Gamma)}= {1} .


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