14.67 (1999)

Suppose that $a$ is a non-trivial element of a finite group $G$ such that $\lvert C_G(a) \rvert \geqslant \lvert C_G(x) \rvert$ for every non-trivial element $x \in G$, and let $H$ be a nilpotent subgroup of $G$ which is normalized by $C_G(a)$. Is it true that $H \leqslant C_G(a)$? This is true if $H$ is abelian, or if $H$ is a $p'$-group for some prime number $p \in \pi(Z(C_G(a)))$.