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

From Groupprops

## Statement

Suppose is a prime number. Consider the map from the category of finite groups to the category of fusion systems that sends a finite group to the fusion system induced on its p-Sylow subgroup. This map is functorial. In particular:

- A homomorphism of groups induces a morphism of fusion systems.
- The morphism of fusion systems induced by the identity morphism is the identity morphism.
- The morphism of fusion systems induced by a composite of two homomorphisms is the composite of the morphisms induced by each of them.