P-nilpotent implies p-normal

Statement
Suppose $$G$$ is a finite group that is a p-nilpotent group, i.e., it has a normal p-complement, or equivalently, the $$p$$-Sylow subgroup is a retract. Then, $$G$$ is a p-normal group: the center of the Sylow subgroup is a weakly closed subgroup in it.

Facts used

 * 1) uses::Retract implies CEP, uses::CEP implies every relatively normal subgroup is weakly closed
 * 2) uses::Center is normal

Proof
The proof follows directly by combining facts (1) and (2).