A note on the order of the Schur multiplier of p-groups

Let G be a finite p-group of order pn with |G′| = pk, and let M(G) denote its Schur multiplier. A classical result of Green states that |M(G)| ≤ p 1 2 n(n−1) . In 2009, Niroomand, improving Green’s and other bounds on |M(G)| for a non-abelian p-group G, proved that |M(G)| ≤ p 2 (n−k−1)(n+k−2)+1. In this paper, we prove that a bound, obtained earlier by Ellis and Wiegold, is stronger than that of Niroomand. We derive from the bound of Ellis and Wiegold that |M(G)| ≤ p 1 2 (d(G)−1)(n+k−2)+1 for a non-abelian p-group G. We obtain an improvement to an old bound given by Vermani. Finally we prove, for a p-group of coclass r, that |M(G)| ≤ p 1 2 (r−r)+kr+1. This improves a bound by Moravec.