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.