Free Abelian Group - Properties

Properties

1. 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 example may be constructed as the abelian group of functions on B, where each function may take integer values, and all but finitely many of its values are zero. This is the direct sum of copies of, one copy for each element of B.
2. 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.
3. 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.
4. 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.
5. Free abelian groups are a special case of free modules, as abelian groups are nothing but modules over the ring .

Importantly, every subgroup of a free abelian group is free abelian (see below). 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.

It can be surprisingly difficult to determine whether a concretely given group is free abelian. Consider for instance the Baer–Specker group, the direct product (not to be confused with the direct sum, which differs from the direct product on an infinite number of summands) of countably many copies of . Reinhold Baer proved in 1937 that this group is not free abelian; Specker proved in 1950 that every countable subgroup of is free abelian.