Generating Countable Sets of Permutations

Abstract

Let E be an infinite set. In answer to a question of Wagon, I show that every countable subset of the symmetric group Sym(E) is contained in a 2-generator subgroup of Sym(E). In answer to a question of Macpherson and Neumann, I show that, if Sym(E) is generated by A ∪ B where |B| ⩽ ‖E‖, then Sym(E) is generated by A ∪ {γ} for some permutation γ in Sym(E).