Free abelian group of rank two

Definition
This group, denoted $$\mathbb{Z}^2$$ or $$\mathbb{Z} \times \mathbb{Z}$$, is defined in the following equivalent ways:


 * 1) It is the free abelian group of rank two.
 * 2) It is the external direct product of two copies of the group of integers.
 * 3) It is the abelianization of free group:F2.