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

From Groupprops

## Definition

Suppose is a group (usually finite, though not necessarily so). Suppose is a subgroup of that is a finite p-group. The fusion system on induced via conjugation by , which we will denote , is a category defined as follows: For any and such that , there is a morphism given by .

Note that this category is a fusion system in the weak sense, but not necessarily a saturated fusion system, which is what many people mean when they talk of fusion system.

## Relation between the transporter system and the fusion system

Suppose is a finite group, is a prime number, and is a finite -subgroup of . Consider the following two categories:

- The transporter system
- The fusion system

The object sets of the two categories are identical, but the morphism sets differ. There is a natural "forgetful" functor from to defined as follows:

- Each object of , namely, a subgroup of , is sent to the same subgroup of , now viewed as an object of .
- The morphism set map is as follows: the element , which is a morphism from to in , gets sent to the homomorphism given by .

## Facts

- When is a finite group and is a -Sylow subgroup of , the category we obtain is a fusion system. This is termed the fusion system induced by a finite group on its Sylow subgroup.
`For full proof, refer: Fusion system induced by a finite group on its p-Sylow subgroup is a saturated fusion system`