On the size of the Schur multiplier of finite groups

TL;DR

This paper provides improved bounds on the size of the Schur multiplier and second cohomology group for finite p-groups, with constructions showing the bounds are sharp in certain cases.

Contribution

It introduces new bounds for the Schur multiplier of finite p-groups, improving upon all previous bounds, and constructs examples achieving these bounds.

Findings

New upper bounds for the Schur multiplier size.

Bounds depend on group parameters like order, derived subgroup size, and generators.

Constructed p-groups that attain the bounds.

Abstract

We obtain bounds for the size of the Schur multiplier of finite $p$-groups and finite groups, which improve all existing bounds. Moreover, we obtain bounds for the size of the second cohomology group $H^2(G,\mathbb{Z}/p\mathbb{Z})$ of a $p$-group with coefficients in $\mathbb{Z}/p\mathbb{Z}$. Denoting the minimal number of generators of a $p$-group $G$ by $d(G)$, our bound depends on the parameters $|G|=p^n$, $|\gamma_2G|=p^k$, $d(G)=d$, $d(G/Z)=\delta$ and $d(\gamma_2G/\gamma_3G)=k'$. For special $p$-groups, we further improve our bound when $\delta-1 > k'$. Moreover, given natural numbers $d$, $\delta$, $k$ and $k'$ satisfying $k=k'$ and $\delta-1 \leq k'$, we construct a capable $p$-group $H$ of nilpotency class two and exponent $p$ such that the size of the Schur multiplier attains our bound.