# 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. A typical example of a free abelian group is the direct sum **Z** ⊕ **Z** of two copies of the infinite cyclic group **Z**; a basis is {(1,0),(0,1)}. The trivial abelian group {0} is also considered to be free abelian, with basis the empty set.

Note a point on terminology: a free abelian group is not the same as a free group that is abelian; in fact the only free groups that are abelian are those of rank 0 (the trivial group) and rank 1 (the infinite cyclic group).

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 **Z**, one copy for each element of *B*.
**Formal sums** of elements of a given set *B* are nothing but the elements of the free abelian group with basis *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.

Given any abelian group *A*, there always exists a free abelian group *F* and a surjective group homomorphism from *F* to *A*. This follows from the universal property mentioned above.

Importantly, every subgroup of a free abelian group is free abelian. As a consequence, to every abelian group *A* there exists a short exact sequence

- 0 →
*G*→*F*→*A*→ 0

with *F* and *G* being free abelian (which means that *A* is isomorphic to the factor group *F*/*G*). This is called a **free resolution** of *A*. Furthermore, the free abelian groups are precisely the projective objects in the category of abelian groups.

All free abelian groups are torsion-free, and all finitely generated torsion-free abelian groups are free abelian. (The same applies to flatness, since an abelian group is torsion-free if and only if it is flat.) The additive group of rational numbers **Q** is a (not finitely generated) torsion-free group that's not free abelian. The reason: **Q** is divisible but non-zero free abelian groups are never divisible.

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

It can be surprisingly difficult to determine whether a concretely given group is free abelian. Consider for instance the Baer-Specker group **Z**^{N}, the direct product of countably many copies of **Z**. R. Baer proved in 1937 that this group is *not* free abelian; Specker proved in 1950 that every countable subgroup of **Z**^{N} is free abelian.