Cubes of conjugacy classes covering the infinite symmetric group

Abstract

Using combinatorial methods, we prove the following theorem on the group S S of all permutations of a countably-infinte set: Whenever p ∈ S p \in S has infinite support without being a fixed-point-free involution, then any s ∈ S s \in S is a product of three conjugates of p p . Furthermore, we present uncountably many new conjugacy classes C C of S S satisfying that any s ∈ S s \in S is a product of two elements of C C . Similar results are shown for permutations of uncountable sets.