Fusion system induced by a finite group on a finite p-subgroup

Definition
Suppose $$G$$ is a group (usually finite, though not necessarily so). Suppose $$P$$ is a subgroup of $$G$$ that is a finite p-group. The fusion system on $$P$$ induced via conjugation by $$G$$, which we will denote $$\mathcal{F}_P(G)$$, is a category defined as follows: For any $$g \in G$$ and $$R,S \le P$$ such that $$gRg^{-1} \le S$$, there is a morphism $$\varphi:R \to S$$ given by $$\varphi(r) = grg^{-1}$$.

Note that this category is a fusion system in the weak sense, but not necessarily a saturated fusion system, which is what many people mean when they talk of fusion system.

Relation between the transporter system and the fusion system
Suppose $$G$$ is a finite group, $$p$$ is a prime number, and $$P$$ is a finite $$p$$-subgroup of $$G$$. Consider the following two categories:


 * The transporter system $$\mathcal{T}_P(G)$$
 * The fusion system $$\mathcal{F}_P(G)$$

The object sets of the two categories are identical, but the morphism sets differ. There is a natural "forgetful" functor from $$\mathcal{T}_P(G)$$ to $$\mathcal{F}_P(G)$$ defined as follows:


 * Each object of $$\mathcal{T}_P(G)$$, namely, a subgroup of $$P$$, is sent to the same subgroup of $$P$$, now viewed as an object of $$\mathcal{F}_P(G)$$.
 * The morphism set map is as follows: the element $$g \in N_G(R,S)$$, which is a morphism from $$R$$ to $$S$$ in $$\mathcal{T}_P(G)$$, gets sent to the homomorphism $$\varphi:R \to S$$ given by $$x \mapsto gxg^{-1}$$.

Facts

 * When $$G$$ is a finite group and $$P$$ is a $$p$$-Sylow subgroup of $$G$$, the category we obtain is a fusion system. This is termed the fusion system induced by a finite group on its Sylow subgroup.