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

TL;DR

This paper refines bounds on the order of the Schur multiplier of finite p-groups, improving previous results and providing sharper estimates based on group structure and coclass.

Contribution

It demonstrates that Ellis and Weigold's bound is more general than Niroomand's, sharpens this bound, and offers new bounds for groups with non-homocyclic abelianization and specific coclass.

Findings

Ellis and Weigold's bound is more general than Niroomand's.

Sharpened bounds for groups with non-homocyclic abelianization.

Improved bounds for p-groups of given coclass.

Abstract

Let $G$ be a finite $p$-group of order $p^n$ with $|G'| = p^k$. Let $M(G)$ denotes the Schur multiplier of $G$. A classical result of Green states that $|M(G)| \leq p^{\frac{1}{2}n(n-1)}$. In 2009, Niroomand, improving Green's and other bounds on $|M(G)|$ for a non-abelain $p$-group $G$, proved that $|M(G)| \leq p^{\frac{1}{2}(n-k-1)(n+k-2)+1}$. In this article we note that a bound, obtained earlier, by Ellis and Weigold is more general than the bound of Niroomand. We derive from the bound of Ellis and Weigold that $|M(G)| \leq p^{\frac{1}{2}(d(G)-1)(n+k-2)+1}$ for a non-abelain $p$-group $G$. Moreover, we sharpen the bound of Ellis and Weigold and as a consequence derive that if $G^{ab}$ is not homocyclic then $|M(G)| \leq p^{\frac{1}{2}(d(G)-1)(n+k-3)+1}$. We further note an improvement in an old bound given by Vermani. Finally we note, for a $p$-group of coclass $r$, that $|M(G)| \leq p^{\frac{1}{2}(r^2-r)+kr+1}$. This improves a bound by Moravec.