Glauberman-Solomon normal p-complement theorem

Statement
Suppose $$G$$ is a finite group and $$p$$ is an odd prime number. Let $$P$$ be a $$p$$-Sylow subgroup of $$G$$. Denote by $$D^*(P)$$ the D*-subgroup of $$P$$. Then, if $$N_G(D^*(P))$$ possesses a normal p-complement (i.e., is a p-nilpotent group), then $$G$$ also possess a normal p-complement (i.e., $$G$$ is also a p-nilpotent group).

In other words, the D*-subgroup functor is a characteristic p-functor that controls normal p-complements.

Facts used

 * 1) uses::Generalized Glauberman-Thompson normal p-complement theorem
 * 2) uses::Strongly p-solvable implies normalizer of D*-subgroup generates whole group with p'-core for odd p

Proof
The proof follows directly by combining Facts (1) and (2). Fact (2) basically tells us that the D*-subgroup satisfies the necessary conditions to apply Fact (1) to it.