Some structural results on the non-abelian tensor square of groups

TL;DR

This paper investigates the structure of the non-abelian tensor square of certain groups, providing conditions for its decomposition and characterizations in terms of presentations, with applications to specific group classes.

Contribution

It offers new structural results on the non-abelian tensor square for groups finitely generated modulo their derived subgroup, including conditions for decomposition and presentation-based characterizations.

Findings

Conditions for $G ensor G$ decomposition into $ abla(G)$ and $G igwedge G$

Characterization of $G igwedge G$ via group presentations

Applications to free soluble, free nilpotent, and finite p-groups

Abstract

We study the non-abelian tensor square $G\otimes G$ for the class of groups G that are finitely generated modulo their derived subgroup. In particular, we find conditions on G/G' so that $G\otimes G$ is isomorphic to the direct product of $\nabla(G)$ and the non-abelian exterior square $G\wedge G$. For any group G, we characterize the non-abelian exterior square $G\wedge G$ in terms of a presentation of G. Finally, we apply our results to some classes of groups, such as the classes of free soluble and free nilpotent groups of finite rank, and some classes of finite p-groups.