Non-abelian tensor product of residually finite groups

Let $G$ and $H$ be groups that act compatibly on each other. We denote by $\eta(G,H)$ a certain extension of the non-abelian tensor product $G \otimes H$ by $G \times H$. Suppose that $G$ is residually finite and the subgroup $[G,H] = \langle g^{-1}g^h \ \mid g \in G, h\in H\rangle$ satisfies some non-trivial identity $f \equiv~1$. We prove that if $p$ is a prime and every tensor has $p$-power order, then the non-abelian tensor product $G \otimes H$ is locally finite. Further, we show that if $n$ is a positive integer and every tensor is left $n$-Engel in $\eta(G,H)$, then the non-abelian tensor product $G \otimes H$ is locally nilpotent. The content of this paper extend some results concerning the non-abelian tensor square $G \otimes G$.Comment: Dedicated to Professor Antonio Paques on the occasion of his 70th anniversary, S\~ao Paulo J. Math. Sci. (2017

A criterion for metanilpotency of a finite group

We prove that the $k$th term of the lower central series of a finite group $G$ is nilpotent if and only if $|ab|=|a||b|$ for any $\gamma_k$-commutators $a,b\in G$ of coprime orders