Squares of conjugacy classes in the infinite symmetric groups

Abstract

Using combinatorial methods, we will examine squares of conjugacy classes in the symmetric groups S ν {S_\nu } of all permutations of an infinite set of cardinality ℵ ν {\aleph _\nu } . For arbitrary permutations p ∈ S ν p \in {S_\nu } , we will characterize when each element s ∈ S ν s \in {S_\nu } with finite support can be written as a product of two conjugates of p p , and if p p has infinitely many fixed points, we determine when all elements of S ν {S_\nu } are products of two conjugates of p p . Classical group-theoretical theorems are obtained from similar results.