# Hall is transitive

From Groupprops

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

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 transitive subgroup property

## Statement

### Verbal statement

Any Hall subgroup of a Hall subgroup of a finite group, is a Hall subgroup in the whole group.

## Facts used

## Proof

**Given**: A finite group , subgroups such that is a Hall subgroup of and is a Hall subgroup of .

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

**Proof**: By fact (1), we have:

.

Now, since is Hall in , the order of is relatively prime to .

By fact (2), the order of divides the order of , and since is a Hall subgroup of ,the order of is relatively prime to . Thus, the order of is relatively prime to .

Thus, the order of is relatively prime to the product , which, by the above equation, equals .