Finitely generated free abelian group

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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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Definition

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

  1. It is isomorphic to a finite direct power of the group of integers, i.e., it is isomorphic to the group \mathbb{Z}^n for some positive integer n.
  2. It is the free abelian group on a finite generating set.
  3. It is the abelianization of a finitely generated free group.
  4. 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