Finitely generated free abelian group
This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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VIEW 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 | |FULL LIST, MORE INFO | |||
| finitely generated nilpotent group | |FULL LIST, MORE INFO |