# Characteristic subgroup of center

From Groupprops

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Contents

## Definition

A subgroup of a group is termed a **characteristic subgroup of center** if is a central subgroup of and is a characteristic subgroup *of* 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 abelian group | the whole group is an abelian group and the subgroup is a characteristic subgroup. | any abelian group is its own center. | |FULL LIST, MORE INFO |

### Weaker properties

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

characteristic central subgroup | the subgroup is a central subgroup and is a characteristic subgroup of the whole group but not necessarily of the center. |
follows from center is characteristic and characteristicity is transitive. | |FULL LIST, MORE INFO | |

powering-invariant characteristic subgroup | characteristic subgroup of center is powering-invariant | |||

quotient-powering-invariant characteristic subgroup | characteristic subgroup of center is quotient-powering-invariant | |||

powering-invariant normal subgroup | ||||

quotient-powering-invariant subgroup | ||||

powering-invariant subgroup | ||||

characteristic subgroup | ||||

central subgroup | ||||

characteristic central factor | ||||

characteristic transitively normal subgroup | ||||

abelian characteristic subgroup | ||||

abelian normal subgroup |