P-normal group

Short version
Suppose $$G$$ is a finite group and $$p$$ is a prime number. We say that $$G$$ is $$p$$-normal if the defining ingredient::conjugacy functor on $$G$$ arising from the defining ingredient::characteristic p-functor sending a finite p-group to its defining ingredient::center is a defining ingredient::weakly closed conjugacy functor on $$G$$.

Long version
Suppose $$G$$ is a finite group and $$p$$ is a prime number. We say that $$G$$ is $$p$$-normal if it satisfies the following equivalent conditions:


 * 1) Either of these equivalent:
 * 2) * There exists a $$p$$-Sylow subgroup $$P$$ of $$G$$ such that the defining ingredient::center $$Z(P)$$ is a defining ingredient::weakly closed subgroup of $$P$$ relative to $$G$$.
 * 3) * For every $$p$$-Sylow subgroup $$P$$ of $$G$$, the center $$Z(P)$$ is a weakly closed subgroup of $$P$$ relative to $$G$$.
 * 4) Either of these equivalent:
 * 5) * There exists a $$p$$-Sylow subgroup $$P$$ such that, for every $$p$$-Sylow subgroup $$Q$$ containing the center $$Z(P)$$, $$Z(P) = Z(Q)$$.
 * 6) * For every $$p$$-Sylow subgroup $$P$$, and for every $$p$$-Sylow subgroup $$Q$$ containing the center $$Z(P)$$, $$Z(P) = Z(Q)$$.
 * 7) Either of these equivalent:
 * 8) * There exists a $$p$$-Sylow subgroup $$P$$ of $$G$$ with center $$Z(P)$$ such that for any $$p$$-Sylow subgroup $$Q$$ of $$G$$ containing the center $$Z(P)$$, $$Z(P)$$ is a normal subgroup of $$Q$$.
 * 9) * For every $$p$$-Sylow subgroup $$P$$ of $$G$$ with center $$Z(P)$$, it is true that for any $$p$$-Sylow subgroup $$Q$$ of $$G$$ containing the center $$Z(P)$$, $$Z(P)$$ is a normal subgroup of $$Q$$.

Equivalence of definitions
The equivalence between both versions of (1), the equivalence between both versions of (2), and the equivalence between both versions of (3), follow from the fact that Sylow implies order-conjugate: any two $$p$$-Sylow subgroups are conjugate, and the conjugating automorphism preserves all properties including weak closure. The (1) implies (2) implies (3) direction is straightforward. The (3) implies (1) direction follows from the fact that characteristic subgroup of Sylow subgroup is weakly closed iff it is normal in every Sylow subgroup containing it.

Incomparable properties

 * p-solvable group: See p-solvable not implies p-normal and p-normal not implies p-solvable