Relative cyclic subgroup commutativity degrees of finite groups

TL;DR

This paper introduces the concept of relative cyclic subgroup commutativity degrees in finite groups, explores their possible values, and demonstrates that the set of all such degrees is not dense in the interval [0,1].

Contribution

It defines a new measure for finite groups, characterizes groups with specific numbers of these degrees, and shows the non-density of their set in [0,1].

Findings

Existence of finite groups with any number of relative cyclic subgroup commutativity degrees except two.

Identification of classes of groups with few such degrees.

The set of all cyclic subgroup commutativity degrees is not dense in [0,1].

Abstract

In this paper we introduce and study the relative cyclic subgroup commutativity degrees of a finite group. We show that there is a finite group with $n$ such degrees for all $n \in \mathbb{N}^*\setminus \lbrace 2\rbrace$ and we indicate some classes of finite groups with few relative cyclic subgroup commutativity degrees. Using this new concept, we are able to show that the set containing all cyclic subgroup commutativity degrees of finite groups is not dense in $[0, 1]$.