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

Abstract

Abstract Let H be a nilpotent subgroup of a finite nonabelian group G, let π = π ⁢ ( | H | ) {\pi=\pi(|H|)} and let bcl ⁡ ( G ) {{\operatorname{bcl}}(G)} be the largest conjugacy class size of the group G. In the present paper, we show that | H ⁢ O π ⁢ ( G ) / O π ⁢ ( G ) | < bcl ⁡ ( G ) {|HO_{\pi}(G)/O_{\pi}(G)|<{\operatorname{bcl}}(G)} .