Free abelian group

Definition
A free abelian group is an abelian group satisfying the following equivalent conditions:

The rank of a free abelian group is defined as the cardinality of a freely generating set for it. The rank of a free abelian group is fixed; in other words, any two freely generating sets of a free abelian group have the same cardinality.

The free abelian group of rank $$\alpha$$ is isomorphic to a direct sum of $$\alpha$$ copies of the group of integers. In particular, the free abelian group of rank $$n$$ for a natural number $$n$$ is isomorphic to the group $$\mathbb{Z}^n$$, which is a direct product of $$n$$ copies of the group of integers.

Extreme examples

 * The trivial group is a free abelian group of rank $$0$$.
 * The group of integers is a free abelian group of rank $$1$$.

Non-examples
An (unrestricted) external direct power of the group of integers need not be free abelian. In fact, it is not free abelian if it is an infinite power. The smallest counterexample is the Baer-Specker group, which arises as the external direct product of countably many copies of the group of integers.