I'm stuck with this problem.
Let $H$ be a normal subgroup of a finite group $G$ such that $C_G(x)\subseteq H$ for every non-identity element $x\in H$ (that is, $H$ is a normal CC-subgroup of $G$). Then $\textrm{gcd}(|H|,|G/H|)=1$ (i.e. $H$ is a Hall-subgroup of $G$).
I think that Sylow's Theorems would be helpful here. I spent hours thinking and I couldn't solve it.
Thanks in advance.