Abstract

Let G be a finite p  -group of order pn, Green proved that M(G), its Schur multiplier is of order at most . 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-abelianp  -group of order pn with derived subgroup of order pk, then . In particular, , and the equality holds in this last bound if and only ifG=H×Z, where H   is extra special of order p³ and exponent p, and Z is an elementary abelian p-group.