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 $xy\equiv zu(\mathrm{mod}n)$, 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 $2$.