Free abelian group

In abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Unlike vector spaces, not all abelian groups have a basis, hence the special name for those that do.

Note a point on terminology: a free abelian group is not the same as a free group that is abelian; in fact most free groups are not abelian.

If F is a free abelian group with basis B, then we have the following universal property: for every arbitrary function f from B to some abelian group A, there exists a unique group homomorphism from F to A which extends f. This universal property can also be used to define free abelian groups.

For every set B, there exists a free abelian group with basis B, and all such free abelian groups having B as basis are isomorphic. One exemplar may be constructed as the abelian group of functions on B, taking integer values all but finitely many of which are zero. This is the direct sum of copies of the infinite cyclic group Z, one copy for each element of B.

Every finitely generated free abelian group is therefore isomorphic to Zⁿ for some natural number n called the rank of the free abelian group. In general, a free abelian group F has many different bases, but all bases have the same cardinality, and this cardinality is called the rank of F. This rank of free abelian groups can be used to define the rank of all other abelian groups: see rank of an abelian group.

Importantly, every subgroup of a free abelian group is free abelian.

Free abelian groups are a special case of free modules, as abelian groups are nothing but modules over the ring Z.