Conjugacy-closed and Sylow implies retract

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 $p$.
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 $p$-Sylow subgroup of a finite group $G$:

1. $P$ is a retract of $G$: there exists a normal complement $N$ to $P$ in $G$.
2. $P$ is conjugacy-closed in $G$: any two elements of $P$ that are conjugate in $G$ are conjugate in $P$.

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

Related facts

For Hall subgroups

• Conjugacy-closed and Hall not implies retract

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.