Free abelian group

Definition

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

1. (Direct sum): It is isomorphic to a possibly infinite restricted direct product (also called direct sum) of copies of the group of integers.
2. (Freely generating set): There exists a subset of the group such that every element of the group has a unique expression as an integer linear combination of elements of the subset. Such a subset is termed a freely generating set or a basis.
3. (Abelianization of free group): It occurs as the abelianization of a free group, i.e., as the quotient of a free group by its commutator subgroup. The image of a freely generating set for the free group forms a basis forms a basis for the free abelian group.

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 generating sets of a free abelian group have the same cardinality. Further information: free abelian groups satisfy IBN