Generating Countable Sets of Permutations

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 answerto a question of Macpherson and Neumann, I show that, if Sym(E)is generated by A B where |B| ||E||, then Sym(E) is generatedby A {} for some permutation in Sym(E).