# Sylow implies order-dominated

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., Sylow subgroup) must also satisfy the second subgroup property (i.e., order-dominated subgroup)

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## Statement

### Statement with symbols

Suppose is a finite group and is a -Sylow subgroup of . Suppose is a subgroup of such that the order of is a multiple of the order of (equivalently, the index of is relatively prime to ). Then, there exists such that .

## Facts used

- Sylow subgroups exist
- Sylow implies order-conjugate: Any two -Sylow subgroups in a finite group are conjugate.

## Proof

**Given**: A finite group , a -Sylow subgroup of . A subgroup of whose order is a multiple of the order of .

**To prove**: There exists such that .

**Proof**:

- has a -Sylow subgroup, say : This follows from fact (1).
- is also a -Sylow subgroup of : This follows from order considerations. Since the order of divides the order of , the largest power of dividing the order of is the same as the largest power of dividing the order of . Thus, a -Sylow subgroup of has the correct order for being a -Sylow subgroup of .
- There exists such that . In particular, : This follows from fact (2), and the previous step, which established that both and are -sylow in .