Conjugacy classes and union of cosets of normal subgroups

Abstract Let G be a finite group, N normal subgroup of and K conjugacy class . We prove that if $$K\cup K^{-1}$$ K ∪ - 1 is union cosets , then soluble, real-imaginary class, is, every irreducible character takes real or purely imaginary values on if, in addition, the elements are p -elements for some prime has -complement. also single coset $$\langle K\rangle $$ ⟨ ⟩ 2-complement.