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

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Definition

Suppose G is a finite group, p is a prime number, and P is a p-Sylow subgroup. The fusion system on P induced by G is defined as follows: for every element 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 fusion system is often written as \mathcal{F}_P(G). The special case where G = P gives rise to what we call the inner fusion system.

This satisfies the conditions necessary for being a saturated fusion system. For full proof, refer: Fusion system induced by a finite group on its p-Sylow subgroup is a saturated fusion system

Note that we can define in a similar way the fusion system induced by a finite group on a finite p-subgroup. However, there is no guarantee in general that this category is a fusion system.

Facts