Non-abelian tensor square of finite-by-nilpotent groups

TL;DR

This paper investigates the properties of the non-abelian tensor square and an extension of finite-by-nilpotent groups, establishing their structural characteristics and providing a characterization of BFC-groups.

Contribution

It proves that the non-abelian tensor square of finite-by-nilpotent groups is also finite-by-nilpotent and characterizes BFC-groups via the extension $ u(G)$.

Findings

Non-abelian tensor square of finite-by-nilpotent groups is finite-by-nilpotent.

The extension $ u(G)$ is nilpotent-by-finite.

Characterization of BFC-groups in terms of $ u(G)$.

Abstract

Let $G$ be a group. We denote by $\nu(G)$ an extension of the non-abelian tensor square $G \otimes G$ by $G \times G$. We prove that if $G$ is finite-by-nilpotent, then the non-abelian tensor square $G \otimes G$ is finite-by-nilpotent. Moreover, $\nu(G)$ is nilpotent-by-finite (Theorem A). Also we characterize BFC-groups in terms of $\nu(G)$ (Theorem B).