Products of conjugacy classes of the infinite symmetric groups

Abstract

Using combinatorial methods, we will examine products of conjugacy classes in the symmetric group S 0 of all permutations of a countably infinite set. If p ϵ S 0 has at least one infinite orbit in the underlying set and if s ϵ S 0, we give a characterization of when s is a product of two conjugates of p. From this, we derive that if four permutations p i ϵ S 0 ( i = 1, 2, 3, 4) are given which all have infinite support, then any permutation of S 0 is a product of four elements conjugate to p 1, p 2, p 3 and p 4, respectively. Similar results for permutations of uncountable sets are shown and classical group-theoretical results are obtained from these theorems.