# Fully invariant subgroup of abelian group

From Groupprops

This article describes a property that arises as the conjunction of a subgroup property: fully invariant 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 **fully invariant subgroup of abelian group** is a fully invariant subgroup of an abelian group, i.e., a subgroup of abelian group that is invariant under all the endomorphisms of the whole group.

## Relation with other properties

### Stronger properties

property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
---|---|---|---|---|

Verbal subgroup of abelian group |

### Weaker properties

property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
---|---|---|---|---|

Characteristic subgroup of abelian group | ||||

Abelian-extensible endomorphism-invariant subgroup | ||||

Abelian-quotient-pullbackable endomorphism-invariant subgroup | ||||

Subgroup of abelian group |