On the Schur multiplier of $p$-groups with abelianization $s$-elementary abelian

TL;DR

This paper presents a method to compute the Schur multiplier of certain finite p-groups with abelianization as an s-elementary abelian group, extending previous work for the case s=1.

Contribution

It generalizes existing methods to compute Schur multipliers for a broader class of p-groups and introduces the concept of s-special p-groups.

Findings

Developed a method to compute Schur multipliers for these p-groups.

Identified which abelian p-groups can be Schur multipliers of non-abelian p-groups.

Computed Schur multipliers for s-special p-groups of rank 1.

Abstract

Let $p$ be an odd prime. We describe a method to compute the Schur multiplier of finite $p$-groups $G$ of nilpotency class $2$ such that $G/[G,G]$ is isomorphic to direct product of copies of $\mathbb{Z}_{p^s}$ for $s \in \mathbb{N}$, generalizing a method of Blackburn and Evens, who treated the case $s=1$. As an application, we investigate which abelian $p$-groups can occur as the Schur multiplier of a non-abelian $p$-group. We further introduce the notions of $s$-special $p$-groups of rank $k$ generalizing the notion of special $p$-groups of rank $k$. We study the structural properties, compute the Schur multipliers of $s$-special $p$-groups of rank $1$.