The Commutativity Degree of \(2\)-Generated Groups of Nilpotency Class \(2\)

Abstract

Let \(G\) be a finite group. The commutativity degree of \(G\), written \(d(G)\), is defined as the ratio \[\frac{|\{(x, y)x,y \in G, xy = yx\}|}{|G|^2}\]. In this paper, we examine the commutativity degree of groups of nilpotency class 2 and, by using numerical solutions of the equation \(xy \equiv zu \pmod{n}\), we give certain explicit formulas for some particular classes of finite groups. A lower bound for \(d(G)\) is obtained for \(2\)-generated groups of nilpotency class \(2\).