A Note on Finite Nilpotent Groups

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.