Grün's second theorem on the focal subgroup

Statement

Suppose $G$ is a finite group and $p$ is a prime number. Suppose further that $G$ is a P-normal group (?): in other words, the center of any $p$-Sylow subgroup is a Weakly closed subgroup of Sylow subgroup (?), i.e., is weakly closed in it. Then, the Focal subgroup (?) of $P$ relative to $G$ equals the focal subgroup of $P$ relative to $N_{G}(Z(P))$.

By the focal subgroup theorem, this is the same as saying that:

$P\cap [G,G]=P\cap [N_{G}(Z(P)),N_{G}(Z(P))]$.

Related facts

• Hall-Wielandt theorem