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