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

Abstract

Let [Formula: see text] be a finite [Formula: see text]-group of order [Formula: see text] with [Formula: see text], and let [Formula: see text] denote its Schur multiplier. A classical result of Green states that [Formula: see text]. In 2009, Niroomand, improving Green’s and other bounds on [Formula: see text] for a non-abelian [Formula: see text]-group [Formula: see text], proved that [Formula: see text]. 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 [Formula: see text] for a non-abelian [Formula: see text]-group [Formula: see text]. We obtain an improvement to an old bound given by Vermani. Finally we prove, for a [Formula: see text]-group of coclass [Formula: see text], that [Formula: see text]. This improves a bound by Moravec.