Some bounds on commutativity degree

TL;DR

This paper investigates bounds and properties of the probability that elements of a subgroup commute with elements of a finite group, exploring invariance under isoclinism and deriving related consequences.

Contribution

It provides new bounds for the relative commutativity degree and studies its invariance under isoclinism of group pairs.

Findings

Established new lower and upper bounds for (H, G)

Analyzed invariance of (H, G) under isoclinism

Derived consequences of the bounds and invariance properties

Abstract

The relative commutativity degree of a subgroup $H$ of a finite group $G$, denoted by $\Pr(H, G)$, is the probability that an element of $G$ commutes with an element of $H$. In this article we obtain some lower and upper bounds for $\Pr(H, G)$ and their consequences. We also study an invariance property of $\Pr(H, G)$ and its generalizations, under isoclinism of pairs of groups.