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It is well known that if G is a group and H is a normal subgroup of G of finite index k, then xᵏ H for every x G. We examine finite groups G with the property that xᵏ H for every subgroup H of G, where k is the index of H in G. We prove that a finite group G satisfies this property if and only if G is nilpotent.
Nicholas J. Werner (Sun,) studied this question.