Commuting Probability of Finite Groups (Extended)

TL;DR

This paper explores the commuting probability in finite groups, providing calculations for specific classes and establishing bounds to measure how close groups are to being abelian.

Contribution

It introduces methods to compute commuting probabilities for various non-abelian groups and derives bounds, advancing understanding of group structure.

Findings

Computed commuting probabilities for several classes of non-abelian groups

Established local and global bounds for commuting probability

Provided insights into the structure of finite groups based on commuting probability

Abstract

The target of this article is to discuss the concept of \textit{commuting probability} of finite groups which, in short, is a probabilistic measure of how abelian our group is. We shall compute the value of commuting probability for many special classes of non-abelian groups and also establish some local and global bounds. We will conclude with a few topics for further reading.