# Abelian normal Hall implies permutably complemented

From Groupprops

## 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 is a finite group and is an abelian normal Hall subgroup of . Then is a permutably complemented subgroup of : there exists a subgroup of such that is trivial and .

## 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.

## Proof

### Proof using group cohomology

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