The exponent of the non-abelian tensor square and related constructions of $p$-groups

Abstract

Let $G$ be a finite $p$-group. In this paper we obtain bounds for the exponent of the non-abelian tensor square $G\otimes G$ and of $\nu (G)$, which is a certain extension of $G\otimes G$ by $G\times G$. In particular, we bound $\exp (\nu (G))$ in terms of $\exp (\nu (G/N))$ and $\exp (N)$ when $G$ admits some specific normal subgroup $N$. We also establish bounds for $\exp (G\otimes G)$ in terms of $\exp (G)$ and either the nilpotency class or the coclass of the group $G$, improving some existing bounds.