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

Abstract

Let G be a finite group. Given two conjugacy classes C and D of G, we shall say that C commutes withD, and write C ∼ D, if there exist elements c ∈ C and d ∈ D such that c and d commute. In this paper we are particularly concerned with the case where G has a normal subgroup H such that G/H is cyclic; in this case, since G/H is abelian, each conjugacy class of G is entirely contained within a particular coset of H. We establish some results concerning the distribution between the cosets of H of pairs of conjugacy classes of G related by ∼. In order to state our theorems, we make the following definition: a conjugacy class gG of G is non-split if gG = gH , and split otherwise. Observe that gG is non-split if and only if the centre CentG(g) of G meets every coset of H in G, or equivalently, if and only if g commutes with an element in a generating coset of G/H. In particular, if a coset Ht generates the quotient group G/H, then all of the conjugacy classes in Ht are non-split.