# Normal Sylow subgroup

From Groupprops

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: normal subgroup and Sylow subgroup

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## Contents

## Definition

A subgroup of a finite group is termed a **normal Sylow subgroup** if it satisfies the following equivalent conditions:

- It is a Sylow subgroup, and is normal in the whole group.
- It is a Sylow subgroup, and is subnormal in the whole group.
- It is a Sylow subgroup, and is characteristic in the whole group.
- It is a Sylow subgroup, and is fully characteristic in the whole group.

## Examples

VIEW: subgroups satisfying this property | subgroups dissatisfying property normal subgroup | subgroups dissatisfying property Sylow subgroupVIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions

## Relation with other properties

### Stronger properties

### Weaker properties

- Nilpotent normal subgroup
- Nilpotent characteristic subgroup
- Normal Hall subgroup
- Complemented normal subgroup
- Fully characteristic subgroup
- Characteristic subgroup
- Intermediately characteristic subgroup
- Isomorph-free subgroup
- Intermediately fully characteristic subgroup
- Image-closed characteristic subgroup
- Image-closed fully characteristic subgroup
- Normal subgroup

## Metaproperties

### Transfer condition

YES:This subgroup property satisfies the transfer condition: if a subgroup has the property in the whole group, its intersection with any subgroup has the property in that subgroup.

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### Intermediate subgroup condition

YES:This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup conditionABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

### Image condition

YES:This subgroup property satisfies the image condition, i.e., under any surjective homomorphism, the image of a subgroup satisfying the property also satisfies the property

View other subgroup properties satisfying image condition