On some series of a group related to the non-abelian tensor square of groups

TL;DR

This paper investigates the structure of a group extension related to the non-abelian tensor square, revealing how its derived subgroup decomposes and describing the lower central series.

Contribution

It introduces a new analysis of the group extension nu(G), showing its derived subgroup as a central product of three isomorphic subgroups, advancing understanding of its structure.

Findings

Derived subgroup nu(G)' is a central product of three isomorphic subgroups.

Structure of each term in the derived and lower central series of nu(G) is characterized.

Provides a detailed description of the extension related to the non-abelian tensor square.

Abstract

Let $G$ be a group. We denote by $\nu(G)$ a certain extension of the non-abelian tensor square $G \otimes G$ by $G \times G$. In this paper we prove that the derived subgroup $\nu(G)'$ is a central product of three normal subgroups of $\nu(G)$, all isomorphic to the non-abelian tensor square $G \otimes G$. As a consequence, we describe the structure of each term of the derived and lower central series of the group $\nu(G)$.