Burnside's normal p-complement theorem
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 result is termed Burnside's normal p-complement theorem and is also sometimes termed Burnside's transfer theorem.
Statement with symbols
- Conjugacy-closed Abelian Sylow implies retract
- Frobenius' normal p-complement theorem
- Thompson's normal p-complement theorem
- Center of Sylow sugbroup is conjugacy-determined in normalizer: If is a Sylow subgroup of , then two elements of are conjugate in if and only if they are conjugate in .
- Conjugacy-closed abelian Sylow implies retract: If is an Sylow subgroup of such that no two distinct elements of are conjugate in , then is a retract of . (Note that the proof of this relies in turn on the focal subgroup theorem).
- Grün's first theorem on the focal subgroup
Proof using a rather weak fusion result
Given: a finite group, a -Sylow subgroup such that .
To prove: is a retract of : it possesses a normal -complement.
- No two distinct elements of are conjugate in : Since is Abelian, , so fact (1) tells us that two elements of are conjugate in if and only if they are conjugate in . Since , no two distinct elements of are conjugate in , and hence no two distinct elements of are conjugate in .
- has a normal complement: This follows from fact (2), since the previous step shows that the conditions for it are satisfied.
Proof using a stronger fusion result
This proof uses fact (3).