UC Berkeley

Let S = Sym(Ω) be the group of all permutations of a countably infinite set Ω, and for subgroups G ₁, G ₂ ≤ S let us write G ₁ ~ G ₂ if there exists a finite set U ⊆ S such that <G ₁ ∪ U> = <G ₂ ∪ 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 ⊆ Ω holds:

(i) For every finite set Γ, the subgroup G _(Γ) has at least one infinite orbit in Ω.

(ii) There exist finite sets Γ such 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 Γ such 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 Γ such that G _(Γ) = {1}.

Some related results and topics for further investigation are noted.