On the structure of normal subgroups of potent p-groups

Let G be a finite p-group satisfying [G,G] G4 for p=2 and γp−1(G) Gp for p>2. The main goal of this paper is to show that any normal subgroup N of G lying in G2 is power abelian, that is, the following holds: (1) Npk={gpk g N}; (2) Ωk(N)={g N o(g) pk}, and (3) Npk=N:Ωk(N).