Burnside's normal p-complement theorem

Name
This result is termed Burnside's normal p-complement theorem and is also sometimes termed Burnside's transfer theorem.

Statement with symbols
Suppose $$p$$ is a prime, $$G$$ is a finite group, and $$P$$ is a $$p$$-fact about::Sylow subgroup of $$G$$. Further, suppose $$P$$ is a fact about::central subgroup of normalizer: if $$H = N_G(P)$$ is its normalizer, and $$Z(H)$$ is the center of $$H$$, then $$P \le Z(H)$$.

Then, $$P$$ is a fact about::retract of $$G$$, i.e., there exists a normal p-complement in $$G$$: a normal subgroup $$N$$ such that $$NP = G$$ and $$N \cap P$$ is trivial.

Related facts

 * Conjugacy-closed Abelian Sylow implies retract
 * Frobenius' normal p-complement theorem
 * Thompson's normal p-complement theorem

Facts used

 * 1) uses::Center of Sylow sugbroup is conjugacy-determined in normalizer: If $$P$$ is a Sylow subgroup of $$G$$, then two elements of $$Z(P)$$ are conjugate in $$G$$ if and only if they are conjugate in $$N_G(P)$$.
 * 2) uses::Conjugacy-closed abelian Sylow implies retract: If $$P$$ is an Sylow subgroup of $$G$$ such that no two distinct elements of $$P$$ are conjugate in $$G$$, then $$P$$ is a retract of $$G$$. (Note that the proof of this relies in turn on the uses::focal subgroup theorem).
 * 3) uses::Grün's first theorem on the focal subgroup

Proof using a rather weak fusion result
Given: $$G$$ a finite group, $$P$$ a $$p$$-Sylow subgroup such that $$P \le Z(N_G(P))$$.

To prove: $$P$$ is a retract of $$G$$: it possesses a normal $$p$$-complement.

Proof:


 * 1) No two distinct elements of $$P$$ are conjugate in $$G$$: Since $$P$$ is Abelian, $$Z(P) = P$$, so fact (1) tells us that two elements of $$P$$ are conjugate in $$G$$ if and only if they are conjugate in $$N_G(P)$$. Since $$P \le Z(N_G(P))$$, no two distinct elements of $$P$$ are conjugate in $$N_G(P)$$, and hence no two distinct elements of $$P$$ are conjugate in $$G$$.
 * 2) $$P$$ 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).