# Normal Hall implies permutably complemented

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This article gives the statement and possibly, proof, of an implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., normal Hall subgroup) must also satisfy the second subgroup property (i.e., permutably complemented subgroup)
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## Name

This result is sometimes called Schur's theorem, and is considered a part of the Schur-Zassenhaus theorem, which also asserts that any two permutable complements to a normal Hall subgroup are conjugate.

## Statement

### Statement with symbols

Suppose $N$ is a normal Hall subgroup of a group $G$. Then there exists a subgroup $H \le G$ such that $N \cap H$ is trivial and $NH = G$. Note that this makes $H$ a Hall subgroup as well.