On the size of the Schur multiplier of finite groups

TL;DR

The article advances the understanding of Schur multipliers by deriving a unified, improved bound for $|M(G)|$ of finite groups and $p$-groups in terms of structural parameters $d$, $\delta$, $k$, and $k'$, refining the Ellis–Wiegold framework. A novel construction based on an alternating bilinear map yields a lower bound on the dimension of $\mathrm{Im}(\Psi)$, which translates into explicit exponent bounds for $|M(G)|$ and, via cohomology, for $|H^2(G,\mathbb{Z}/p\mathbb{Z})|$. The authors extend the bounds to all finite groups by Sylow decomposition and demonstrate sharpness through concrete examples and a general constructive method that attains equality for special $p$-groups and their extensions. The results improve previous bounds (e.g., Rai 2024) in many parameter regimes and provide practical constructions of groups attaining the bound, with implications for group capability and cohomology investigations.

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 $|G|=p^n$, $|γ_2G|=p^k$, $d(G)=d$, $d(G/Z)=δ$ and $d(γ_2G/γ_3G)=k'$. For special $p$-groups, we further improve our bound when $δ-1 > k'$. Moreover, given natural numbers $d$, $δ$, $k$ and $k'$ satisfying $k=k'$ and $δ-1 \leq k'$, 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.

Figures (1)

• Figure 1:

Theorems & Definitions (38)

• Theorem 1.1
• Remark 1
• Definition 2.1
• Remark 2
• Lemma 2.2
• proof
• Proposition 2.3
• proof
• Lemma 2.4
• Remark 3