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Let $G$, $H$ be groups. We denote by $\eta(G,H)$ a certain extension of the non-abelian tensor product $G \otimes H$ by $G \times H$. We prove that if $G$ and $H$ are groups that act compatibly on each other and such that the set of all tensors $T_{\otimes}(G,H)=\{g\otimes h \, : \, g \in G, \, h\in H\}$ is finite, then the non-abelian tensor product $G \otimes H$ is finite. In the opposite direction we examine certain finiteness conditions of $G$ in terms of similar conditions for the tensor square $G \otimes G$.Comment: The content of this paper improves and extends the main results of arXiv:1603.07003 [math.GR

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We study the non-abelian tensor square modulo q of a group, where q is a nonnegative integer, via an operator n q in the class of groups. Structural properties and finiteness conditions of n q ðGÞ are investigated. We compute the non-abelian tensor square modulo q of cyclic groups and develop a theory for computing n q ðGÞ and some of its relevant sections for polycyclic groups G. This extends the existing theory from the case q ¼ 0 to all nonnegative integers q. Additionally, a table of examples is produced with the help of the GAP system.