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 $R,S\leq P$ such that $gRg^{-1}\leq S$, 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. For full proof, refer: Fusion system induced by a finite group on its p-Sylow subgroup is a saturated fusion system