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.