Certain Classes of Groups with Commutativity Degree $d(G)<\frac{1}{2}$

Abstract

For a finite group $G$ the commutativity degree,

$$d(G)=\frac{|\{(x,y)|x,y\in G,xy=yx\}|}{|G{|}^2}$$

is defined and studied by several authors and when $d(G)\ge \frac{1}{2}$ it is proved by P. Lescot in 1995 that $G$ is abelian , or $\frac{G}{Z(G)}$ is elementary abelian with $|G^{\prime }|=2$, or $G$ is isoclinic with $S_3$ and $d(G)=1$. The case when $d(G)<\frac{1}{2}$ is of interest to study. In this paper we study certain infinite classes of finite groups and give explicit formulas for $d(G)$. In some cases the groups satisfy $\frac{1}{4}<d(G)<\frac{1}{2}$. Some of the groups under study are nilpotent of high nilpotency classes.