Conjugacy-closed and Sylow implies retract

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.

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.

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.