The order of 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 bounds based on the size of certain related sets and exploring finiteness criteria for tensor squares.

Contribution

It provides new bounds on the order of the non-abelian tensor product based on the size of specific element sets, advancing understanding of its finiteness properties.

Findings

The derivative subgroup [G,H] is finite with m-bounded order when a certain set has m elements.

The non-abelian tensor product G ⊗ H is finite with m-bounded order if the set of tensors has m elements.

Finiteness conditions for the non-abelian tensor square of groups are established.

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.