# Sylow implies order-dominating

From Groupprops

This article gives the statement and possibly, proof, of an implication relation between two subgroup properties, when the big group is a finite group. That is, it states that in a Finite group (?), every subgroup satisfying the first subgroup property (i.e., Sylow subgroup (?)) must also satisfy the second subgroup property (i.e., Order-dominating subgroup (?)). In other words, every Sylow subgroup of finite group is a order-dominating subgroup of finite group.

View all subgroup property implications in finite groups View all subgroup property non-implications in finite groups View all subgroup property implications View all subgroup property non-implications

## Contents

## Statement

### Property-theoretic statement

The subgroup property of being a Sylow subgroup is stronger than, or implies, the subgroup property of being an order-dominating subgroup: for any subgroup whose order divides the order of the Sylow subgroup, some conjugate of that subgroup is contained in the given Sylow subgroup.

### Statement with symbols

Suppose is a finite group and is a prime number. Then, if is a -Sylow subgroup and is a -subgroup of , there exists such that .

This statement is a part of Sylow's theorem.

## Related facts

### Corollaries

- Sylow implies order-conjugate: This follows from the fact that order-dominating implies order-conjugate in co-Hopfian: in particular, in a finite group, any order-dominating subgroup is order-conjugate. In other words, any two -Sylow subgroups are conjugate.

## Proof

### Proof using coset spaces

**Given**: a finite group, a -Sylow subgroup, and a -group. Suppose where is the order of and is relatively prime to (So, .

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

**Proof**: We prove this through a series of observations:

- naturally acts (on the left) on the left coset space of .
- Since is a subgroup of , also acts on the left coset space of
- The left coset space of has cardinality , which is relatively prime to . Hence, under the action of (which is a -group) on this set, there is at least one fixed point. Let the fixed point be .
- We have . This gives us: , and hence . Thus, a conjugate of is contained inside .