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 $x \mapsto gxg^{-1}$.