# Every p'-group is the p'-part of the automorphism group of a p-group

## Statement

Suppose is a prime number and is a finite group whose order is no a multiple of . Then, there exists a -group such that has a normal -Sylow subgroup, with as a complement to the normal Sylow subgroup.

Further, the normal -Sylow subgroup is the kernel of the action of on .

## Facts used

- Cayley's theorem
- Bryant-Kovacs theorem
- Burnside's theorem on coprime automorphisms and Frattini subgroup
- Normal Hall implies permutably complemented

## Proof

**Given**: A finite group , a prime not dividing the order of .

**To prove**: There exists a -group such that has a normal -Sylow subgroup with complement .

**Proof**: There exists for which can be embedded as a subgroup of (this follows from fact (1), and the fact that the symmetric group on letters embeds inside ). By fact (2), we can find a -group such that the image of in is .

By fact (3), the kernel of the map from to is a -group. Since the kernel is a -group and the quotient is a -group, the kernel is a normal -Sylow subgroup. By fact (4), it has a permutable complement in , and this complement is isomorphic to .