# 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 $NH = G$.