Weakly closed subgroup for a fusion system

Definition
Suppose $$P$$ is a group of prime power order and $$\mathcal{F}$$ is a fusion system on $$P$$. Then a subgroup $$Q \le P$$ is termed weakly closed with respect to $$\mathcal{F}$$ if for every morphism $$\varphi: Q \to P$$ in $$\mathcal{F}$$, we have $$\varphi(Q) = Q$$.

Stronger properties

 * Weaker than::Strongly closed subgroup for a fusion system

Related subgroup properties and subgroup-of-subgroup properties

 * Weakly closed subgroup: Suppose $$H \le K \le G$$ are groups. $$H$$ is weakly closed in $$K$$ with respect to $$G$$ if, for any $$g \in G$$, $$gHg^{-1} \le K$$ implies that $$gHg^{-1} \le H$$.
 * Weakly closed subgroup of Sylow subgroup: The case where $$K$$ is a $$p$$-Sylow subgroup of $$G$$. This is related to fusion systems as follows: a subgroup of a $$p$$-Sylow subgroup $$K$$ of a finite group $$G$$ is weakly closed in $$K$$ if and only if it is weakly closed for the fusion system induced by $$G$$ on $$K$$.
 * Sylow-relatively weakly closed subgroup: A subgroup of a group of prime power order that is a weakly closed subgroup in any group containing the bigger group as a Sylow subgroup.
 * Fusion system-relatively weakly closed subgroup: A subgroup of a group of prime power order that is weakly closed in the fusion system sense).for any fusion system on the bigger group.