# Characteristic central factor

From Groupprops

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

View other subgroup property conjunctions | view all subgroup properties

## Contents

## Definition

A subgroup of a group is termed a **characteristic central factor** if it satisfies the following two conditions:

- It is a characteristic subgroup: every automorphism of the whole group restricts to an automorphism of the subgroup.
- It is a central factor: every inner automorphism of the whole group restricts to an inner automorphism of the subgroup.

## Relation with other properties

### Stronger properties

### Weaker properties

- Characteristic subgroup
- Central factor
- Characteristic transitively normal subgroup
- Conjugacy-closed characteristic subgroup
- Conjugacy-closed normal subgroup
- SCAB-subgroup
- Transitively normal subgroup
- Normal subgroup

## Metaproperties

### Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitiveABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity