Normal fusion subsystem

Definition
A fusion subsystem $$\mathcal{G}$$ of a fusion system $$\mathcal{F}$$ on a group of prime power order $$P$$ is termed a normal fusion subsystem if:


 * The subgroup $$Q$$ of $$P$$ for which $$\mathcal{G}$$ is a fusion system is a strongly closed subgroup of $$P$$. In other words, for any $$\varphi:R \to P$$ with $$\varphi \in \mathcal{F}$$, $$\varphi(Q \cap R) \le Q$$.
 * Conjugation of any morphism in $$\mathcal{G}$$ by a morphism in $$\mathcal{F}$$ gives a morphism in $$\mathcal{G}$$, in the following sense: If $$\varphi \in \mathcal{F}$$ and $$\alpha \in \mathcal{G}$$ are morphisms such that $$\varphi \circ \alpha \circ \varphi^{-1}$$ is well-defined and between two objects of $$\mathcal{G}$$ (i.e., two subgroups of $$Q$$), then $$\varphi\circ \alpha \circ \varphi^{-1} \in \mathcal{G}$$.