Finite split metacyclic groups and their 2-nilpotent multipliers

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.