Conjugacy-closed and Sylow implies retract

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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

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: