Cosets of normal subgroups and union of two conjugacy classes

TL;DR

This paper investigates conditions under which cosets of normal subgroups in finite groups are contained within the union of two conjugacy classes, revealing restrictions on the subgroup structure and the nature of simple factors involved.

Contribution

It characterizes when cosets of normal subgroups lie in the union of two conjugacy classes, especially in non-solvable cases involving simple groups of Lie type.

Findings

Cosets can be contained in the union of two conjugacy classes even if the normal subgroup is non-abelian simple.

Such conjugacy classes must have equal cardinality in these cases.

Non-solvable structure of the normal subgroup is constrained to products of simple groups of Lie type.

Abstract

Let $G$ be a finite group, $N$ a normal subgroup of $G$ and $x\in G-N$. We discuss when the coset $Nx$ is contained in the union of two conjugacy classes, $K$ and $D$, of $G$. We show that $N$ need not be solvable, and can even be non-abelian simple, but in these cases, $K$ and $D$ must have the same cardinality, and the non-solvable structure of $N$ is restricted. The non-abelian principal factors of $G$ contained in $N$ are then isomorphic to $S\times \cdots\times S$, where $S$ is a simple group of Lie type of odd characteristic.