# Normal fusion subsystem

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 $\varphi\circ \alpha \circ \varphi^{-1} \in \mathcal{G}$.