# Hall satisfies intermediate subgroup condition

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., Hall subgroup) satisfying a subgroup metaproperty (i.e., intermediate subgroup condition)

View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties

Get more facts about Hall subgroup |Get facts that use property satisfaction of Hall subgroup | Get facts that use property satisfaction of Hall subgroup|Get more facts about intermediate subgroup condition

## Contents

## Statement

Any Hall subgroup of a finite group is also a Hall subgroup in any intermediate subgroup.

## Related facts

- Sylow satisfies intermediate subgroup condition: Essentially, the same proof.

## Facts used

## Proof

**Given**: A finite group , a Hall subgroup , and a subgroup of containing .

**To prove**: is a Hall subgroup of .

**Proof**: Note that by the multiplicativity of index:

.

Thus, the index divides the index . In particular, if and are relatively prime, so are and .