Non-abelian tensor product of residually finite groups

Abstract

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 extends some results concerning the non-abelian tensor square $G \otimes G$.