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

Abstract

Let $G$ be a group. We denote by $\nu (G)$ a certain extension of the non-abelian tensor square $[G,G^{\phi }]$ by $G\times G$. We prove that if $G$ is a finite potent $p$-group, then $[G,G^{\phi }]$ 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).