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 interesting 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.