Conjugacy class numbers and nilpotent subgroups of finite groups (Q6634497)

Let \(G\) be a finite group, \(k(G)\) the number of conjugacy classes of \(G\) and \(\mathrm{acs}(G)=|G|/k(G)\) the average conjugacy class size of \(G\).\N\NLet \(B \leq G\) be a nilpotent subgroup of \(G\) and \(\pi=\pi(G)\). The main results in the paper under review are: (a) if \(G\) is solvable, then \(|B O_{\pi}(G)/O_{\pi}(G)| \leq \mathrm{acs}(G)\); (b) if \(G\) is non-solvable, then \(|B O_{\pi}(G)/O_{\pi}(G)| \leq \frac{7}{15}\mathrm{acs}(G)\).