Permutable subgroups of a direct product

A subgroup S of a group G is a permutable subgroup of G if for all subgroups X of G, SX=XS. In this article, we establish necessary and sufficient conditions for a subgroup of the finite group G×H to be permutable. Then, we attempt to improve this theorem by making conjectures that would simplify these conditions. Counterexamples to these conjectures are presented, demonstrating that in some way, the aforementioned characterization is the best one possible. We conclude by showing how our conjectures do provide further insight into permutability in some special cases.