# Self-centralizing normal subgroup

From Groupprops

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

View other subgroup property conjunctions | view all subgroup properties

## Contents

## Definition

### Symbol-free definition

A subgroup of a group is termed a **self-centralizing normal subgroup** or **normal self-centralizing subgroup** if it satisfies the following equivalent conditions:

- It is normal and self-centralizing in the whole group (i.e., it contains its centralizer in the whole group).
- The induced map from the quotient of the whole group by its center, to the automorphism group of the subgroup, is injective.

### Definition with symbols

A subgroup of a group is termed a self-centralizing normal subgroup if it satisfies the following equivalent conditions:

- and .
- The induced homomorphism , via the action by conjugation, is injective.

## Relation with other properties

### Stronger properties

### Weaker properties

- Coprime automorphism-faithful subgroup (for finite groups):
`For full proof, refer: Self-centralizing and normal implies coprime automorphism-faithful` - Normality-large normal subgroup and Normality-large subgroup:
`For full proof, refer: Self-centralizing and normal implies normality-large`

## Metaproperties

### 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