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 \(d(G) \geq \frac{1}{2}\) it is proved by P. Lescot in 1995 that \(G\) is abelian , or \(\frac{G}{Z(G)}\) is elementary abelian with \(|G’| = 2\), 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.