The Non-Abelian Tensor Square of Finite Metacyclic Groups
Juliana Silva Canella, Norai Romeu Rocco
TL;DR
The paper addresses the problem of determining explicit presentations for the non-abelian tensor square $G \otimes G$ of finite metacyclic groups $G=g(a,b;m,n,r,s)$ with odd $m$, by leveraging the auxiliary group $\nu(G)$ and its commutator-based structure. It develops a concrete, commutator-driven framework to describe $\nu(G)$ and its relevant sections, including the non-abelian exterior square $G \wedge G$ and the Schur multiplier $M(G)$, via canonical central and kernel relations. For $m$ odd, it provides an explicit finite presentation of $\nu(G)$ through a group $M$, proves $\nu(G) \cong M$, and derives that $G \otimes G$ is abelian with a concrete presentation on generators $u,v,w,z$; it also yields presentations for $G \wedge G$ and $M(G)$. A corollary specializes to split metacyclic groups, delivering detailed descriptions of $\nu(G)$, $G \otimes G$, $G \wedge G$, and $M(G)$ in that case, and the work aligns with and extends earlier biderivation-based approaches and known multipliers.
Abstract
In this paper, we investigate the group $ν(G)$, an extension of the non-abelian tensor square $G$ by the direct product $G\times G$, in order to determine a presentation of $G \otimes G$ when $G$ is a general finite metacyclic group, $G=g(a,b; m,n, r, s)$, with $m$ odd. A presentation of $ν(G)$ is obtained from that of $G$ and, consequently, we describe the appropriate relevant sections of $ν(G)$, such as the non-abelian tensor square, the exterior square $G \wedge G$ and the Schur multiplier, $M(G)$.