Non-abelian tensor product of residually finite groups

TL;DR

This paper investigates conditions under which the non-abelian tensor product of residually finite groups is locally finite or locally nilpotent, extending previous results on the non-abelian tensor square.

Contribution

It establishes new criteria involving identities and Engel conditions that ensure local finiteness or nilpotency of the non-abelian tensor product of residually finite groups.

Findings

If all tensors have p-power order, then the tensor product is locally finite.

If tensors are left n-Engel, then the tensor product is locally nilpotent.

Results extend known properties of the non-abelian tensor square G ⊗ G.

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