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

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## Definition

Suppose is a finite group, is a prime number, and is a -Sylow subgroup. The fusion system on induced by is defined as follows: for every element , and subgroups such that , there is a morphism given by .

This fusion system is often written as . The special case where 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.