On the distribution of conjugacy classes between the cosets of a finite group in a cyclic extension

TL;DR

This paper investigates how conjugacy classes are distributed across cosets in finite groups with cyclic extensions, providing elementary proofs and applications to counting conjugacy classes.

Contribution

It establishes that conjugacy classes with a specific centralizing subgroup are evenly distributed among cosets in certain group extensions, with new elementary proof techniques.

Findings

Conjugacy classes are equally distributed among cosets in cyclic extensions.

Provides formulas for counting conjugacy classes based on group actions.

Elementary proof approach simplifies understanding of class distribution.

Abstract

Let G be a finite group and H a normal subgroup such that G/H is cyclic. Given a conjugacy class g^G of G we define its centralizing subgroup to be HC_G(g). Let K be such that H\le K\le G. We show that the G-conjugacy classes contained in K whose centralizing subgroup is K, are equally distributed between the cosets of H in K. The proof of this result is entirely elementary. As an application we find expressions for the number of conjugacy classes of K under its own action, in terms of quantities relating only to the action of G.