# Normal fully normalized 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 fully normalized subgroup

View other subgroup property conjunctions | view all subgroup properties

## Contents

## Definition

### Symbol-free definition

A subgroup of a group is termed **normal fully normalized** if it is normal in the whole group, and every automorphism of the subgroup extends to an inner automorphism of the whole group.

### Definition with symbols

A subgroup of a group is termed **normal fully normalized** if is normal in , and every can be realized as conjugation by for some .

Equivalently, is normal in , and the natural map is surjective.

## Relation with other properties

### Conjunction with other properties

- NSCFN-subgroup is the conjunction with the property of being a self-centralizing subgroup.

### Weaker properties

## Facts

- Every group is normal fully normalized in its holomorph
- Normal upper-hook fully normalized implies characteristic
- Characteristically simple and normal fully normalized implies minimal normal

## Metaproperties

### Left realization

Every group can be realized as a normal fully normalized subgroup in some group, for instance, in its holomorph. `Further information: Every group is normal fully normalized in its holomorph`