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

Commuting elements in conjugacy classes: An application of Hall's Marriage Theorem