Free abelian group of rank two

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VIEW DEFINITIONS USING THIS AS EXAMPLE: All terms |Group properties |Subgroup properties|
VIEW FACTS USING THIS AS EXAMPLE: All facts |Group property non-implications |Subgroup property non-implications |Group metaproperty dissatisfactionsSubgroup metaproperty dissatisfactions

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.

Group properties

Property	Meaning	Satisfied?	Explanation
abelian group	any two elements commute	Yes
torsion-free group	no nonzero elements of finite order	Yes
finitely generated group	has a finite generating set	Yes