Fusion system induced by a finite group on its p-Sylow subgroup is the inner fusion system iff the group is p-nilpotent

Statement
Suppose $$G$$ is a finite group and $$P$$ is a $$p$$-Sylow subgroup of $$G$$. Then, the fusion system induced by $$G$$ on $$P$$ (i.e., $$\mathcal{F}_P(G)$$) equals the inner fusion system $$\mathcal{F}_P(P)$$ if and only if $$G$$ is a fact about::p-nilpotent group, i.e., $$G$$ contains a fact about::normal p-complement.

Facts used

 * 1) uses::Conjugacy-closed and Sylow implies retract

Proof
Given: A finite group $$G$$, a $$p$$-Sylow subgroup $$P$$ of $$G$$, such that $$\mathcal{F}_P(G) = \mathcal{F}_P(P)$$.

To prove: $$G$$ has a normal $$p$$-complement.

Proof:


 * 1) Suppose $$a,b \in P$$ are conjugate in $$G$$. Then, $$a,b$$ are conjugate in $$P$$: Let $$Q = \langle a \rangle, R = \langle b \rangle$$. Then, if $$gag^{-1} = b$$ consider the map $$\varphi:Q \to R$$ given by $$\varphi(x) = gxg^{-1}$$. This is a morphism in $$\mathcal{F}_P(G)$$, so it is also a morphism in $$\mathcal{F}_P(P)$$. In particular, there exists $$h \in P$$ such that $$hah^{-1} = b$$.
 * 2) $$P$$ has a normal complement in $$G$$, so $$G$$ has a normal $$p$$-complement: This follows from the previous step and fact (1).