Finiteness conditions for the non-abelian tensor product of groups

Abstract

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

Finiteness of homotopy groups related to the non-abelian tensor product

“…Specifically, our proofs involve looking at the description of the diagonal subgroup (G) [G, G ϕ ]. Such a description has previously been used by the authors [2,21].…”

“…In [18], Parvizi and Niroomand prove that if G is a finitely generated group and the non-abelian tensor square [G, G ϕ ] is finite, then G is finite. Later, in [2], the authors prove that if G is a finitely generated locally graded group and the exponent of the non-abelian tensor square exp([G, G ϕ ]) is finite, then G is finite. The next result can be viewed as a generalization of the above results.…”

“…In [12], Ellis gives a finiteness criterion for the triad homotopy group in terms of the finiteness of the involved groups (see also [2,3]). More generally, finiteness conditions on π n (X ) when the excision theorem holds are given by the finiteness of π n (A), π n (B) and π n−1 (C).…”

“…For the convenience of the reader we repeat the relevant definitions (cf. [2,3,16]). Let G and H be groups each of which acts upon the other (on the right),…”

“…When G = H and all actions are by conjugation, we simply write T ⊗ (G) instead of T ⊗ (G, G). A number of structural results for the non-abelian tensor product of groups (and related constructions) in terms of the set of tensors were presented in [2][3][4][5]21].…”