Nilpotency of Normal Subgroups Having Two G-class Sizes

Abstract

Let G be a finite group. If N is a normal subgroup which has exactly two G-conjugacy class sizes, then N is nilpotent. In particular, we show that N is abelian or is the product of a p-group P by a central subgroup of G. Furthermore, when P is not abelian, P/(Z(G) ∩ P ) has exponent p.