# Sylow satisfies permuting transfer condition

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

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## Contents

## Statement

### Statement with symbols

Suppose is a Sylow subgroup of a finite group . Suppose, further, that is a subgroup of such that and are permuting subgroups -- in other words, . Then, is a Sylow subgroup of .

## Definitions used

### Sylow subgroup

`Further information: Sylow subgroup`

A subgroup of a finite group is termed a **Sylow subgroup** if its order and its index are relatively prime.

## Facts used

- Index is multiplicative
- Lagrange's theorem
- Product formula: This states that if and are subgroups of , we have:

.

## Proof

**Given**: A finite group , a Sylow subgroup of , a subgroup of such that .

**To prove**: is Hall in .

**Proof**: Rearranging the product formula (fact (3)) yields:

.

By Lagrange's theorem (fact (2)), and noting that is a subgroup of , we get:

.

By fact (1), we have:

.

Thus, we get:

.

In particular, divides . By Lagrange's theorem, we have that divides . Since and are relatively prime, we obtain that and are relatively prime. Thus, is a Sylow subgroup of .