Let G be a finite p-group of order pn, Green proved that M (G), its Schur multiplier is of order at most pfrac(1, 2) n (n - 1). Later Berkovich showed that the equality holds if and only if G is elementary abelian of order pn. In the present paper, we prove that if G is a non-abelian p-group of order pn with derived subgroup of order pk, then | M (G) | ≤ pfrac(1, 2) (n + k - 2) (n - k - 1) + 1. In particular, | M (G) | ≤ pfrac(1, 2) (n - 1) (n - 2) + 1, and the equality holds in this last bound if and only if G = H × Z, where H is extra special of order p3 and exponent p, and Z is an elementary abelian p-group. © 2009 Elsevier Inc. All rights reserved.
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