  Let G be a finite group. Define a relation ~ on the conjugacy classes of G by
setting 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, 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.
