On a connection between the order of a finite group and the set of conjugacy classes size

Abstract

Let $G\in \{p,q{\}}^{\ast }$ be a finite group with trivial center, where $p,q\in \pi (G)$ and $p>q>5$. In the present paper it is proved that $|G{|}_{\{p,q\}}=|G|{|}_{\{p,q\}}$; in particular $C_G(g)\cap C_G(h)=1$ for every $p$-element $g$ and every $q$-element $h$.