Considerations on the subgroup commutativity degree and related notions

TL;DR

This paper explores a stronger version of the subgroup commutativity degree in finite groups, analyzing its properties and relationships with the traditional commutativity degree.

Contribution

It introduces and studies a new, stronger notion of subgroup commutativity degree and establishes its connections with existing measures.

Findings

A new stronger subgroup commutativity degree is defined.

Relations between the new measure and the classical commutativity degree are established.

The paper provides insights into the probability of subgroup pairs commuting.

Abstract

The concept of subgroup commutativity degree of a finite group $G$ is arising interest in several areas of group theory in the last years, since it gives a measure of the probability that a randomly picked pair $(H,K)$ of subgroups of $G$ satisfies the condition $HK=KH$. In this paper, a stronger notion is studied and relations with the commutativity degree are found.