The order of the non-abelian tensor product of groups

Abstract

Let $G$ and $H$ be groups that act compatibly on each other. We denote by $[G,H]$ the derivative subgroup of $G$ under $H$. We prove that if the set $\{g^{-1}g^h \mid g \in G, h \in H\}$ has $m$ elements, then the derivative $[G,H]$ is finite with $m$-bounded order. Moreover, we show that if the set of all tensors $T_{\otimes}(G,H) = \{g\otimes h \mid g \in G, h\in H\}$ has $m$ elements, then the non-abelian tensor product $G \otimes H$ is finite with $m$-bounded order. We also examine some finiteness conditions for the non-abelian tensor square of groups.