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 $[G,H]=⟨g^{-1}{g}^h\mid g\in G,h\in H⟩$ 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 $G\otimes G$.

Dedicated to Professor Antonio Paques on the occasion of his 70th anniversary, São Paulo J. Math. Sci. (2017)

References in corpus (1)

• The non-abelian tensor square of residually finite groups

Cited by in corpus (3)

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• Finiteness conditions for the weak commutativity construction
• The order of the non-abelian tensor product of groups