Abelian normal Hall implies permutably complemented

Name
This result is sometimes termed Schur's theorem. It was originally proved by Schur. Schur and Zassenhaus later extended this to prove that any normal Hall subgroup (not necessarily an abelian one) is permutably complemented.

Statement
Suppose $$G$$ is a finite group and $$N$$ is an abelian normal Hall subgroup of $$G$$. Then $$N$$ is a permutably complemented subgroup of $$G$$: there exists a subgroup $$H$$ of $$G$$ such that $$N \cap H$$ is trivial and $$NH = G$$.

Related facts

 * Normal Hall implies permutably complemented: This is a slight generalization that drops the abelianness assumption. The proof of the general case works by reducing it to the abelian case.