Cosets of normal subgroups and union of two conjugacy classes

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.