Closed subgroups of the infinite symmetric group

Abstract

Let S=Sym(\Omega) be the group of all permutations of a countably infinite set \Omega, 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_{(\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\}. Some questions for further investigation are discussed.