# Sylow implies pronormal

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

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## 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).