Finitely generated free abelian group

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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