# Fusion system induced by a finite group on its p-Sylow subgroup is the inner fusion system iff the group is p-nilpotent

From Groupprops

## Statement

Suppose is a finite group and is a -Sylow subgroup of . Then, the fusion system induced by on (i.e., ) equals the inner fusion system if and only if is a P-nilpotent group (?), i.e., contains a Normal p-complement (?).

## Facts used

## Proof

**Given**: A finite group , a -Sylow subgroup of , such that .

**To prove**: has a normal -complement.

**Proof**:

- Suppose are conjugate in . Then, are conjugate in : Let . Then, if consider the map given by . This is a morphism in , so it is also a morphism in . In particular, there exists such that .
- has a normal complement in , so has a normal -complement: This follows from the previous step and fact (1).