The largest lengths of conjugacy classes and the Sylow subgroups of finite groups. (Q818718)

From the authors' summary: Let \(G\) be a finite nonabelian group, \(P\in\text{Syl}_p(G)\), and \(\text{bcl}(G)\) the largest length of conjugacy classes of \(G\). In this short paper, we prove that \(\sqrt{|P/O_p(G)|}<\text{bcl}(G)\) in general and \(|P/O_p(G)|<\text{bcl}(G)\) in the case where \(P\) is Abelian.