On the structure of normal subgroups of potent p-groups

Let G be a finite p-group satisfying [G, G] less than or equal to G(4) for p = 2 and gamma(p- 1) (G) less than or equal to G(p) for p > 2 The main goal of this paper is to show that any normal subgroup N of G lying in G(2) is power abelian, that is, the following holds: (1) N-pk = {g(pk) \ g is an element of N}; (2) Omega(k)(N) = {g is an element of N \ o(g) less than or equal to p(k)}, and (3) \N-pk\ = \N : Omega(k)(N)\. (C) 2004 Elsevier Inc. All rights reserved.