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 **Z**^{n} 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**.