Glauberman-Thompson normal p-complement theorem

Direct statement
Suppose $$G$$ is a finite group and $$p$$ is an odd prime number. Let $$P$$ be a $$p$$-Sylow subgroup. Then, if $$N_G(Z(J(P))$$ possesses a normal p-complement, so does $$G$$. In other words, $$P$$ is a retract of $$G$$.

In terms of functors and control of complements
For an odd prime $$p$$, the ZJ-functor is a characteristic p-functor that  controls normal p-complements in every finite group.

Related facts

 * Burnside's normal p-complement theorem
 * Frobenius' normal p-complement theorem
 * Thompson's first normal p-complement theorem
 * Thompson's second normal p-complement theorem
 * Glauberman-Solomon normal p-complement theorem

Facts used

 * 1) uses::Generalized Glauberman-Thompson normal p-complement theorem
 * 2) uses::Strongly p-solvable implies Glauberman type for odd p: This states that in a strongly p-solvable group, the ZJ-subgroup functor is a characteristic p-functor whose normalizer generates whole group with p'-core. In particular, for a p'-core-free strongly p-solvable group, the ZJ-subgroup of any p-Sylow subgroup is normal (in fact, characteristic) in the whole group. See also group of Glauberman type for a prime.

Proof
The proof follows directly from Facts (1) and (2). Fact (2) basically says that the ZJ-subgroup functor satisfies the necessary conditions for us to be able to apply Fact (1).