# Conjugacy-closed and Sylow implies retract

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

This article gives a proof/explanation of the equivalence of multiple definitions for the term Sylow retract

View a complete list of pages giving proofs of equivalence of definitions

## Statement

The following are equivalent for a -Sylow subgroup of a finite group :

- is a retract of : there exists a normal complement to in .
- is conjugacy-closed in : any two elements of that are conjugate in are conjugate in .

This result is a part of Frobenius' normal p-complement theorem.

## Related facts

### For Hall subgroups

### Converse

It is in general true that any retract is conjugacy-closed, but the converse is not true. Thus, the condition of being Sylow plays a crucial role here.

`Further information: Retract implies conjugacy-closed, conjugacy-closed not implies retract`

### Weaker facts

These facts are special cases of this general fact, but have easier and less intensive proofs:

- Conjugacy-closed Abelian Sylow implies retract: The special case when the Sylow subgroup is Abelian. This is also a weak formulation of Burnside's normal p-complement theorem.
- Equivalence of definitions of Sylow direct factor: The special case when the Sylow subgroup is normal.