# Finitely generated free abelian group

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Definition

A group is termed a **finitely generated free abelian group** if it satisfies the following equivalent conditions:

- It is isomorphic to a finite direct power of the group of integers, i.e., it is isomorphic to the group for some positive integer .
- It is the free abelian group on a finite generating set.
- It is the abelianization of a finitely generated free group.
- It is a finitely generated abelian group as well as a torsion-free abelian group.

## Relation with other properties

### Weaker properties

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

finitely generated abelian group | |FULL LIST, MORE INFO | |||

free abelian group | |FULL LIST, MORE INFO | |||

finitely generated free nilpotent group | |FULL LIST, MORE INFO | |||

finitely generated free solvable group | |FULL LIST, MORE INFO | |||

torsion-free abelian group | Free abelian group|FULL LIST, MORE INFO | |||

finitely generated nilpotent group | Finitely generated abelian group|FULL LIST, MORE INFO |