# Non-abelian tensor square and related constructions of $p$-groups

## TL;DR

This paper investigates the structure of non-abelian tensor squares and related constructions in finite potent and powerful p-groups, establishing embedding properties and exponent bounds.

## Contribution

It introduces new embedding results and exponent divisibility properties for non-abelian tensor squares of potent p-groups.

## Key findings

- Potent embedding of [G,G^φ] in ν(G) for finite potent p-groups
- Exponent of ν(G) divides p times the exponent of G for potent p-groups
- Analysis of weak commutativity in powerful p-groups

## Abstract

Let $G$ be a group. We denote by $\nu(G)$ a certain extension of the non-abelian tensor square $[G,G^{\varphi}]$ by $G \times G$. We prove that if $G$ is a finite potent $p$-group, then $[G,G^{\varphi}]$ and the $k$-th term of the lower central series $\gamma_k(\nu(G))$ are potently embedded in $\nu(G)$ (Theorem A). Moreover, we show that if $G$ is a potent $p$-group, then the exponent $\exp(\nu(G))$ divides $p \cdot \exp(G)$ (Theorem B). We also study the weak commutativity construction of powerful $p$-groups (Theorem C).