Squares of conjugacy classes in the infinite symmetric groups

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