Cyclic subgroup commutativity degrees of finite groups

TL;DR

This paper introduces the cyclic subgroup commutativity degree, a measure of how likely two random cyclic subgroups of a finite group commute, providing formulas for specific groups and criteria for Iwasawa groups.

Contribution

It defines and analyzes the cyclic subgroup commutativity degree, offering explicit formulas and a criterion for identifying Iwasawa groups.

Findings

Explicit formulas for certain classes of groups

A criterion for a finite group to be an Iwasawa group

Probability measure of commuting cyclic subgroups

Abstract

In this paper we introduce and study the concept of cyclic subgroup commutativity degree of a finite group $G$. This quantity measures the probability of two random cyclic subgroups of $G$ commuting. Explicit formulas are obtained for some particular classes of groups. A criterion for a finite group to be an Iwasawa group is also presented.