# Fusion system induced by a finite group on its p-Sylow subgroup is a saturated fusion system

## Statement

Suppose $G$ is a finite group, $p$ is a prime number, and $P$ is a $p$-Sylow subgroup of $G$. Suppose $\mathcal{F}$ is defined as a category with objects the subgroups of $P$ and morphisms given as follows: For every $g \in G$ and subgroups $R,S \le P$ such that $gRg^{-1} \le S$, there is a morphism $\varphi:R \to S$ given by $\varphi(r) = grg^{-1}$. This is a category on $P$ and is in fact a fusion system.

## Proof

### Proof that it is a category

First, we prove that we get a category on $P$.

• It is closed under composition: Suppose $\alpha:R \to S$ and $\beta:S \to T$ are morphisms induced by conjugation by $g,h \in G$ respectively. Then, $gRg^{-1} \le S$ and $hSh^{-1} \le T$, so $hgRg^{-1}h^{-1} \le T$. Thus, there is a morphism in $\mathcal{F}$ from $R$ to $T$ given by conjugation by $hg$. We see that this is the composite $\beta \circ \alpha$.
• All morphisms are group homomorphisms: This follows because each morphism arises as a restriction of an inner automorphism on the whole group.
• It contains all inclusion maps: This follows because we can always set $g$ to be the identity element.
• If $\varphi:R \to S$ is a morphism, then the restriction $\varphi:R \to \varphi(R)$, and its inverse, are morphisms. This follows from the definition.

### Proof that it satisfies the conditions for a fusion system

We verify the three conditions:

• All morphisms induced by inner automorphisms from $P$ are present: This is true, because in fact all morphisms induced by inner automorphisms from $G$ are present.
• The inner automorphisms of $P$ form a $p$-Sylow subgroup of $\operatorname{Aut}_{\mathcal{F}}(P)$: $\operatorname{Aut}_{\mathcal{F}}(P)$ comprises the automorphisms induced by elements of $N_G(P)$. Since $P$ is $p$-Sylow in $G$, it is $p$-Sylow in $N_G(P)$, so since Sylow satisfies image condition, the image $\operatorname{Aut}_P(P) = \operatorname{Inn}(P)$ is $p$-Sylow in $\operatorname{Aut}_{\mathcal{F}}(P)$.
• The extension axiom: PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]