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

Abstract

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 2 n ( 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 | M ( G ) | ⩽ p 1 2 ( n + k − 2 ) ( n − k − 1 ) + 1 . In particular, | M ( G ) | ⩽ p 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 p 3 and exponent p, and Z is an elementary abelian p-group.