# Difference between revisions of "Sylow implies pronormal"

From Groupprops

(New page: {{subgroup property implication in| group property = finite group| stronger = Sylow subgroup| weaker = pronormal subgroup}} ==Statement== Any Sylow subgroup of a finite group is ...) |
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* [[Sylow normalizer implies abnormal]] | * [[Sylow normalizer implies abnormal]] | ||

+ | * [[Sylow normalizer implies upward-closed self-normalizing]] | ||

* [[Sylow implies intermediately subnormal-to-normal]] | * [[Sylow implies intermediately subnormal-to-normal]] | ||

## Latest revision as of 19:30, 19 September 2008

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., Pronormal subgroup (?)). In other words, every Sylow subgroup of finite group is a pronormal 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

Any Sylow subgroup of a finite group is pronormal.

## Definitions used

### Sylow subgroup

`Further information: Sylow subgroup`

### Pronormal subgroup

`Further information: Pronormal subgroup`

## Related facts

### Stronger facts

### Corollaries

- Sylow normalizer implies abnormal
- Sylow normalizer implies upward-closed self-normalizing
- Sylow implies intermediately subnormal-to-normal

## Facts used

## Proof

The proof follows directly by combining facts (1) and (2).