The largest conjugacy class size and the nilpotent subgroups of finite groups (Q667023)

Let \(G\) be a finite group and \(\mathrm{bcl}(G)\) be the largest conjugacy class size of \(G\). Let \(H \leq G\) be a nilpotent subgroup of \(G\) and let \(\pi=\pi(G)\). In the article under review, the following remarkable result is proved: \[ | HO_{\pi}(G)/O_{\pi}G)| < \mathrm{bcl}(G), \] in particular, if \(P\) is a Sylow \(p\)-subgroup of \(G\), then \(| P/O_{p}(G) | <\mathrm{bcl}(G)\). The proof relies on the classification of finite simple groups and on the description of their largest nilpotent subgroups.