Groups with a conjugacy class that is the difference of two normal subgroups

TL;DR

This paper characterizes finite groups with a conjugacy class equal to the difference of two normal subgroups, providing structural insights and a character-theoretic perspective, generalizing previous results by Gagola.

Contribution

It offers a novel character-theoretic characterization and structural analysis of groups with a conjugacy class as the difference of two normal subgroups, extending Gagola's work.

Findings

Characterization of such conjugacy classes via character theory

Structural properties of groups with these classes

Generalization of Gagola's result when a certain minimal normal subgroup condition is met

Abstract

We consider finite groups having a conjugacy class that is the difference of two normal subgroups. That is, suppose $G$ is a group and $M$ and $N$ are normal subgroups so that $N < M$, and suppose that there is an element $g \in G$ so that the conjugacy class of $g$ is $M \setminus N$. We find a character-theoretic characterization of this condition, and we determine some structural properties of groups with such a conjugacy class. If we add the condition that $M/N$ is the unique minimal normal subgroup of $G/N$, then we obtain a generalization of a result by S.M. Gagola.