Finite groups with two relative subgroup commutativity degrees

TL;DR

This paper investigates the properties of finite groups with specific numbers of relative subgroup commutativity degrees, identifying conditions for multiple degrees and characterizing certain groups like the dihedral group D6.

Contribution

It demonstrates the existence of infinitely many finite groups with exactly two relative subgroup commutativity degrees and characterizes the dihedral group D6 as unique in this regard.

Findings

Infinite groups with two relative subgroup commutativity degrees exist.

D6 is the only finite dihedral group with two such degrees.

Conditions for having at least three degrees are provided.

Abstract

In this paper we show that there is an infinite number of finite groups with two relative subgroup commutativity degrees. Also, we indicate a sufficient condition such that a finite group has at least three relative subgroup commutativity degrees and we prove that $D_6$ is the only finite dihedral group with two relative commutativity degrees. Finally, we study the density of the set containing all subgroup commutativity degrees of finite groups.