Some bounds on commutativity degree

Abstract

The relative commutativity degree of a subgroup $H$ of a finite group $G$, denoted by $\mathrm{Pr}(H,G)$, 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 $\mathrm{Pr}(H,G)$ and their consequences. We also study an invariance property of $\mathrm{Pr}(H,G)$ and its generalizations, under isoclinism of pairs of groups.

10 pages