Finiteness conditions for the non-abelian tensor product of groups

TL;DR

This paper investigates conditions under which the non-abelian tensor product of groups is finite, establishing that finiteness of tensors implies finiteness of the tensor product itself, with implications for group finiteness properties.

Contribution

It proves that finiteness of the set of tensors implies finiteness of the non-abelian tensor product for groups acting compatibly, and explores related finiteness conditions.

Findings

Finiteness of tensors implies finiteness of the tensor product.

Conditions for finiteness of the tensor square $G ensor G$.

Extension $ta(G,H)$ relates to group finiteness.

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 $G \otimes G$.