# Characteristic central subgroup

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 subgroup

View other subgroup property conjunctions | view all subgroup properties

## Contents

## Definition

A subgroup of a group is termed a **characteristic central subgroup** if it satisfies **both** these conditions:

- is a characteristic subgroup of .
- is a central subgroup of , i.e., it is contained in the center of .

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

characteristic subgroup of center | central subgroup that is characteristic as a subgroup of the center. | follows from center is characteristic and characteristicity is transitive | ||

characteristic subgroup of abelian group | Characteristic subgroup of center|FULL LIST, MORE INFO |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

characteristic subgroup | ||||

central subgroup | ||||

characteristic central factor | ||||

characteristic transitively normal subgroup | ||||

abelian characteristic subgroup | ||||

abelian normal subgroup |