Finitely generated free abelian group

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.