On the Tensor Degree of Finite Groups

Abstract

We study the number of elements $x$ and $y$ of a finite group $G$ such that $x\otimes y=1_{G\oplus G}$ in the nonabelian tensor square $G\otimes G$ of $G$. This number, divided by $|G{|}^2$, is called the tensor degree of $G$ and has connections with the exterior degree, introduced a 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.