The non-abelian tensor square of residually finite groups

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 $[x,y^{\phi }{]}^q=1$, then the derived subgroup $\nu (G{)}^{\prime }$ 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 $[x,y^{\phi }{]}^q$ is left $n$-Engel, then the non-abelian tensor square $G\otimes G$ is locally virtually nilpotent (Theorem B).

11 pages. arXiv admin note: substantial text overlap with arXiv:1505.04468

References in corpus (3)

• On profinite groups with Engel-like conditions
• Non-abelian tensor square of finite-by-nilpotent groups
• On residually finite groups with Engel-like conditions

Cited by in corpus (3)

• Finiteness conditions for the non-abelian tensor product of groups
• Finiteness of homotopy groups related to the non-abelian tensor product
• Non-abelian tensor product of residually finite groups