Equivalence of definitions of weakly closed conjugacy functor

Statement
Suppose $$G$$ is a finite group, $$p$$ a prime number, and $$W$$ a defining ingredient::conjugacy functor on $$G$$ with respect to $$p$$. The following are equivalent:


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

Similar facts

 * Characteristic subgroup of Sylow subgroup is weakly closed iff it is normal in every Sylow subgroup containing it

Applications

 * Equivalence of normality and characteristicity conditions for conjugacy functor
 * Equivalence of definitions of characteristic p-functor whose normalizer generates whole group with p'-core

Facts used

 * 1) uses::Sylow implies order-conjugate
 * 2) uses::Sylow implies WNSCDIN (used only in (3) implies (1) proof)
 * 3) uses::Conjugacy functor gives normalizer-relatively normal subgroup (used only in abstract version of (3) implies (1) proof)
 * 4) uses::WNSCDIN implies every normalizer-relatively normal conjugation-invariantly relatively normal subgroup is weakly closed (used only in abstract version of (3) implies (1) proof)

Preliminary notes
The equivalence between both versions of (1), the equivalence between both versions of (2), and the equivalence between both versions of (3), follow from Fact (1) (Sylow implies order-conjugate): any two $$p$$-Sylow subgroups are conjugate, and the conjugating automorphism preserves all properties including weak closure.

(1) implies (2)
Given: A finite group $$G$$, a prime number $$p$$, a $$p$$-conjugacy functor $$W$$ in $$G$$ and $$p$$-Sylow subgroups $$P,Q$$ of $$G$$ such that $$W(P)$$ is weakly closed in $$P$$ (with respect to $$G$$) and also $$W(P)$$ is contained in $$Q$$.

To prove: $$W(P) = W(Q)$$

Proof:

(2) implies (3)
Given: A finite group $$G$$, a prime number $$p$$, a $$p$$-conjugacy functor $$W$$ in $$G$$, $$p$$-Sylow subgroups $$P,Q$$ satisfying $$W(P) = W(Q)$$.

To prove: $$W(P)$$ is normal in $$Q$$.

Proof:

(3) implies (1) (concrete proof)
Given: A finite group $$G$$, a prime number $$p$$, a conjugacy functor $$W$$ and a $$p$$-Sylow subgroup $$P$$ of $$G$$ such that for any $$p$$-Sylow subgroup $$Q$$ of $$G$$ containing $$W(P)$$, $$W(P)$$ is normal in $$Q$$. $$g \in G$$ is such that $$gW(P)g^{-1} \le P$$.

To prove: $$gW(P)g^{-1} = W(P)$$.

Proof:

(3) implies (1) (abstract proof)
Given: Finite group $$G$$, prime $$p$$, $$p$$-conjugacy functor $$W$$, $$p$$-Sylow subgroup $$P$$ of $$G$$. $$W(P)$$ is conjugation-invariantly relatively normal in $$P$$.

To prove: $$W(P)$$ is weakly closed in $$P$$.

Proof: