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

TL;DR

This paper investigates the distribution of conjugacy class pairs in finite groups with cyclic quotients, applying Hall's Marriage Theorem to establish new results and exploring specific cases like symmetric and linear groups.

Contribution

It introduces new theorems on conjugacy class relations in groups with cyclic quotients, utilizing Hall's Marriage Theorem in novel ways.

Findings

Distribution of conjugacy class pairs in groups with cyclic quotients

Application of Hall's Marriage Theorem to group theory

Analysis of symmetric and linear groups

Abstract

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.