A free Abelian group is a group G with a subset which generates the group G with the only relation being a b = b a. That is, it has no group torsion. All such groups are a direct product of the integers Z, and have rank given by the number of copies of Z. For example, Z*Z = {(n, m)} is a free Abelian group of rank 2. A minimal subset b_1, ..., b_n that generates a free Abelian group is called a basis, and gives G as G = Z b_1 + ... + Z b_n. A free Abelian group is an Abelian group, but is not a free group (except when it has rank one, i.e., Z). Free Abelian groups are the free modules in the case when the ring is the ring of integers Z.

Abelian group | free group | free module | free product | group | group torsion