The exponent of non-abelian tensor square of $p$-groups.

Let $G$ be a group. We denote by $\nu(G)$ a certain extension of the non-abelian tensor square $[G,G^{\varphi}]$ by $G \times G$. In this paper we obtain bounds for the exponent of $\nu(G)$, when $G$ is a finite $p$-group. In particular, we prove that if $N$ is a potent normal subgroup of a $G$, then $\exp(\nu(G))$ divides ${\bf p} \cdot \exp(N) \cdot \exp(\nu(G/N))$, where ${\bf p}$ denotes the prime $p$ if $p$ is odd and $4$ if $p=2$. As an application, we show that if $G$ is a $p$-group of maximal class, then $\exp(\nu(G))$ divides ${\bf p}^2 \cdot \exp(G)$. We also establish a bound to $\exp(\nu(G))$ in terms of $\exp(G)$ and the coclass of $G$. Consequently, we improve some previous bounds to the exponent of Schur Multiplier $M(G)$.