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

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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 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).