Finite split metacyclic groups and their 2-nilpotent multipliers

TL;DR

This paper investigates the 2-nilpotent multipliers of finite split metacyclic groups, providing explicit descriptions using their nonabelian tensor squares, which advances understanding of their algebraic structure.

Contribution

It offers a complete characterization of the 2-nilpotent multiplier for finite split metacyclic groups via nonabelian tensor squares, a novel explicit description.

Findings

Explicit formulas for the 2-nilpotent multiplier of the groups.

Descriptions of the triple tensor and exterior products.

Enhanced understanding of the algebraic structure of these groups.

Abstract

There has been a great importance in understanding the nilpotent multipliers of finite groups in recent past. Let a group $G$ be presented as the quotient of a free group $F$ by a normal subgroup $R$. Given a positive integer $c$, the $c$-nilpotent multiplier of the group $G$ is the abelian group $\mathcal M^{(c)}(G)=(R\cap \gamma_{c+1}(F))/\gamma_{c+1}(R,F)$, where $\gamma_1(R,F)=R$, $\gamma_{c+1}(R,F)=[\gamma_c(R,F),F]$, and $\gamma_{c+1}(F)=\gamma_{c+1}(F,F)$. In particular, $\mathcal{M}^{(1)}(G)$ is the Schur multiplier of $G$. The crucial aspect of the research in to the $c$-nilpotent multipliers of groups includes either establishing their structures, or estimating their sizes and exponents. One reason for studying the $c$-nilpotent multiplier is its relevance to the isologism theory of P. Hall. The study of Schur multiplier of finite metacyclic groups goes back to the paper by F. R. Beyl in 1973. In this article, we study the 2-nilpotent multiplier of finite split metacyclic groups with the help of their nonabelian tensor squares. In particular, we give a complete description of the triple tensor product, the triple exterior product, and the 2-nilpotent multiplier of such groups.