On the tensor degree of finite groups

TL;DR

This paper investigates the tensor degree of finite groups, which measures the proportion of element pairs with trivial tensor product in the nonabelian tensor square, revealing structural insights about the groups.

Contribution

It introduces the tensor degree of finite groups, explores bounds for it, and connects these bounds to the structural properties of the groups.

Findings

Established bounds for the tensor degree of finite groups.

Connected tensor degree bounds to structural restrictions of groups.

Related tensor degree to the exterior degree for finite groups.

Abstract

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 $|G|^2$, 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.