The non-abelian tensor square of residually finite groups

TL;DR

This paper investigates the structure of the non-abelian tensor square of residually finite groups, establishing conditions under which certain subgroups are locally finite or virtually nilpotent, based on identities and Engel conditions.

Contribution

It introduces new criteria involving $p$-powers and identities that determine local finiteness and nilpotency properties of the tensor square of residually finite groups.

Findings

The derived subgroup $ u(G)'$ is locally finite under specified conditions.

The non-abelian tensor square $G ensor G$ is locally virtually nilpotent given certain Engel conditions.

Conditions involving $p$-powers and identities influence the group's tensor square structure.

Abstract

Let $m,n$ be positive integers and $p$ a prime. 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 a residually finite group satisfying some non-trivial identity $f \equiv~1$ and for every $x,y \in G$ there exists a $p$-power $q=q(x,y)$ such that $[x,y^{\varphi}]^q = 1$, then the derived subgroup $\nu(G)'$ is locally finite (Theorem A). Moreover, we show that if $G$ is a residually finite group in which for every $x,y \in G$ there exists a $p$-power $q=q(x,y)$ dividing $p^m$ such that $[x,y^{\varphi}]^q$ is left $n$-Engel, then the non-abelian tensor square $G \otimes G$ is locally virtually nilpotent (Theorem B).