# Burnside's normal p-complement theorem

From Groupprops

This article gives the statement, and possibly proof, of a normal p-complement theorem: necessary and/or sufficient conditions for the existence of a Normal p-complement (?). In other words, it gives necessary and/or sufficient conditions for a given finite group to be a P-nilpotent group (?) for some prime number .

View other normal p-complement theorems

## Contents

## Name

This result is termed **Burnside's normal p-complement theorem** and is also sometimes termed **Burnside's transfer theorem**.

## Statement

### Statement with symbols

Suppose is a prime, is a finite group, and is a -Sylow subgroup (?) of . Further, suppose is a Central subgroup of normalizer (?): if is its normalizer, and is the center of , then .

Then, is a Retract (?) of , i.e., there exists a normal p-complement in : a normal subgroup such that and is trivial.

## Related facts

- Conjugacy-closed Abelian Sylow implies retract
- Frobenius' normal p-complement theorem
- Thompson's normal p-complement theorem

## Facts used

- Center of Sylow sugbroup is conjugacy-determined in normalizer: If is a Sylow subgroup of , then two elements of are conjugate in if and only if they are conjugate in .
- Conjugacy-closed abelian Sylow implies retract: If is an Sylow subgroup of such that no two distinct elements of are conjugate in , then is a retract of . (Note that the proof of this relies in turn on the focal subgroup theorem).
- Grün's first theorem on the focal subgroup

## Proof

### Proof using a rather weak fusion result

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

**To prove**: is a retract of : it possesses a normal -complement.

**Proof**:

- No two distinct elements of are conjugate in : Since is Abelian, , so fact (1) tells us that two elements of are conjugate in if and only if they are conjugate in . Since , no two distinct elements of are conjugate in , and hence no two distinct elements of are conjugate in .
- has a normal complement: This follows from fact (2), since the previous step shows that the conditions for it are satisfied.

### Proof using a stronger fusion result

This proof uses fact (3).