# Sylow satisfies intermediate subgroup condition

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

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## Statement

A Sylow subgroup of a finite group is also a Sylow subgroup in any intermediate subgroup.

## Definitions used

### Sylow subgroup

`Further information: Sylow subgroup`

A subgroup of a group is a Sylow subgroup if is a group of prime power order and the order of is relatively prime to the index of .

is termed a -Sylow subgroup if its order is a power of the prime and its index is not a multiple of .

## Related facts

## Facts used

## Proof

**Given**: A finite group , a -Sylow subgroup of , a subgroup of containing .

**To prove**: is a Sylow subgroup (in fact, a -Sylow subgroup) inside .

**Proof**: Clearly, continues to be a -subgroup inside , because it is a -group. Thus, it suffices to show that the index is relatively prime to . For this, observe that by fact (1):

Thus, is a divisor of . Since is relatively prime to , so is .