The largest size of conjugacy class and the $p$-parts of finite groups

Abstract

Let $p$ be a prime and let $P$ be a Sylow $p$-subgroup of a finite nonabelian group $G$. Let $bcl(G)$ be the size of the largest conjugacy class of the group $G$. We show that $|P/O_p(G)|<bcl(G)$ if $G$ is not abelian.