Finitely generated free abelian group
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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
|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|