Commuting conjugacy classes: an application of Hall's marriage theorem to group theory

Let G be a finite group. Define a relation ~ on the conjugacy classes of G by setting C ~ D if there are representatives c C and d D such that cd = dc. In the case where G has a normal subgroup H such that G/H is cyclic, two theorems are proved concerning the distribution, between cosets of H, of pairs of conjugacy classes of G related by ~. One of the proofs involves an application of the famous marriage theorem of Philip Hall. The paper concludes by discussing some aspects of these theorems and of the relation ~ in the particular cases of symmetric and general linear groups, and by mentioning an open question related to Frobenius groups.