On the connection between \(\mu(G)\) and the structure of finite groups. (Q851512)

Let \(G\) be a finite group, \(k(G)\) the number of conjugacy classes of \(G\) and \(\mu(G)=|G|/k(G)\). In this paper, the authors investigate the connection between \(\mu(G)\) and the structure of \(G\). This work is a generalization of related theorems [see \textit{Y. Berkovich}, Proc. Am. Math. Soc. 121, No. 3, 679-685 (1994; Zbl 0820.20006), and \textit{W.-J. Shi} and \textit{Y.-R. Xiao}, Southeast Asian Bull. Math. 22, No. 3, 301-305 (1998; Zbl 0940.20026)].