A description of the derived series and the lower central series of a non-abelian tensor product G n H is given. For a ®nite solvable group G, we obtain an upper bound for the order of G n G.

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 report on a group construction in connection with non-abelian tensor products of groups and recent development in non-abelian tensor products and q-tensor products.