Characterizing finite $p$-groups by their Schur multipliers

TL;DR

This paper classifies the structure of finite p-groups with a specific Schur multiplier parameter t(G)=4, extending previous characterizations for smaller values of t(G) and removing additional conditions.

Contribution

It provides a complete classification of p-groups with t(G)=4 without any extra assumptions, advancing the understanding of their structure.

Findings

Classified p-groups with t(G)=4

Extended previous results for t(G)=0,1,2,3

Removed conditions on the center of the group

Abstract

It has been proved in \cite{ge} for every $p$-group of order $p^n$, $|\mathcal{M}(G)|=p^{\f{1}{2}n(n-1)-t(G)}$, where $t(G)\geq 0$. In \cite{be, el, zh}, the structure of $G$ has been characterized for $t(G)=0,1,2,3$ by several authors. Also in \cite{sa}, the structure of $G$ characterized when $t(G)=4$ and $Z(G)$ is elementary abelian. This paper is devoted to classify the structure of $G$ when $t(G)=4$ without any condition.