Abstract Green [ Proc. Roy. Soc. Math A 327 (1956) , 574–581] proved that if G is a finite p-group of order p n and M( G) is its Schur multiplier of order p m( G) , then m(G) ⩽ 1 2 n(n − 1) . We consider, among other things, the case when m(G) = 1 2 n(n − 1) and then prove that if this equality holds, then G = E( p n ), that is, G is an elementary abelian group of order p n .