# Normal Hall implies permutably complemented

From Groupprops

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

### Verbal statement

Any normal Hall subgroup of a group is permutably complemented.

### Statement with symbols

Suppose is a normal Hall subgroup of a group . Then there exists a subgroup such that is trivial and . Note that this makes a Hall subgroup as well.