On the tensor degree of finite groups

We study the number of elements $x$ and $y$ of a finite group $G$ such that $x\otimes y=1_{{}_{G\otimes G}}$ in the nonabelian tensor square $G\otimes G$ of $G$. This number, divided by $\mathrm{\mid }G{\mathrm{\mid }}^2$, is called the tensor degree of $G$ and has connection with the exterior degree, introduced few years ago in [P. Niroomand and R. Rezaei, On the exterior degree of finite groups, Comm. Algebra 39 (2011), 335–343]. The analysis of upper and lower bounds of the tensor degree allows us to find interesting structural restrictions for the whole group.