Closed subgroups of the infinite symmetric group

Abstract

Let S=Sym(Ω) be the group of all permutations of a countably infinite set Ω, and for subgroups G_1, G_2\leq 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. Which of these classes a closed subgroup G belongs to depends on which of the following statements about pointwise stabilizer subgroups G_{(Γ)} of finite subsets Γ\subseteqΩholds: (i) For every finite set Γ, the subgroup G_{(Γ)} has at least one infinite orbit in Ω. (ii) There exist finite sets Γsuch that all orbits of G_{(Γ)} are finite, but none such that the cardinalities of these orbits have a common finite bound. (iii) There exist finite sets Γsuch that the cardinalities of the orbits of G_{(Γ)} have a common finite bound, but none such that G_{(Γ)}=\{1\}. (iv) There exist finite sets Γsuch that G_{(Γ)}=\{1\}. Some questions for further investigation are discussed.