P-nilpotent implies p-normal

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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. Retract implies CEP, CEP implies every relatively normal subgroup is weakly closed
  2. Center is normal

Proof

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