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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

Nilpotency of normal subgroups having two 𝐺-class sizes