# Subgroup of abelian group

This article describes a property that arises as the conjunction of a subgroup property: subgroup with a group property imposed on theambient group: abelian group

View a complete list of such conjunctions | View a complete list of conjunctions where the group property is imposed on the subgroup

## Contents

## Definition

A subgroup of a group is termed a **subgroup of abelian group** if the ambient group is an abelian group.

## Relation with other properties

### Stronger properties

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

subgroup of cyclic group | the whole group is cyclic | |FULL LIST, MORE INFO |

### Weaker properties

Properties weaker than this are properties true for all subgroups of abelian groups.

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

central subgroup | contained in the center | |FULL LIST, MORE INFO | ||

central factor | product with centralizer is whole group | (via central subgroup) | (via central subgroup) | Central subgroup|FULL LIST, MORE INFO |

transitively normal subgroup | every normal subgroup of it is normal in the whole group. | (via central factor) | (via central factor) | Central subgroup|FULL LIST, MORE INFO |

normal subgroup | abelian implies every subgroup is normal | any non-abelian group as a subgroup of itself. | Central subgroup, Normal subgroup of nilpotent group|FULL LIST, MORE INFO | |

abelian normal subgroup | the subgroup is abelian as a group and is normal in the whole group. | (via central subgroup) | (via central subgroup) | Central subgroup|FULL LIST, MORE INFO |