On the order of Schur multiplier of non-abelian p-groups

Let G be a finite p-group of order p(n), Green proved that M(G). its Schur multiplier is of order at most p(1/2n(n-1)). Later Berkovich showed that the equality holds if and only if G is elementary abelian of order p(n). In the present paper, we prove that if G is a non-abelian p-group of order p(n) with derived subgroup of order p(k), then vertical bar M(G)vertical bar <= p(1/2(n+k-2)(n-k-1)+1). In particular, vertical bar M(G)vertical bar <= p(1/2(n-1)(n-2)+1), and the equality holds in this last bound if and only if G = H x Z, where H is extra special of order p(3) and exponent p, and Z is an elementary abelian p-group. (C) 2009 Elsevier Inc. All rights reserved.