Free Abelian Groups: Isomorphic to Z x Z...xZ?

[ehrenfest]

Homework Statement

What is the point of giving free abelian groups a special name if they are all isomorphic to Z times Z times Z ... times Z for r factors of Z, where r is the rank of the basis?

Homework Equations

The Attempt at a Solution

 

[masnevets]

This is the same as asking why we talk about an n-dimensional vector space (say real) when it is just isomorphic to R^n. The point is that yes, it is isomorphic to R^n, but not in any canonical way. What I mean is that the isomorphism depends on a choice of basis, and therefore is not natural.

Take for example, the set of homomorphisms from Z^2 to Z, denoted Hom(Z^2, Z). This is a free Abelian group that isomorphic to Z^2, but there is no natural isomorphism. (try to find one and you'll see that you keep having to pick a basis of Z^2 to do so)