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

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SUMMARY

Free abelian groups are defined as groups that are isomorphic to the direct sum of r copies of the integers, denoted as Z x Z x ... x Z, where r represents the rank of the basis. The discussion highlights that while free abelian groups are isomorphic to Z^r, the isomorphism is not canonical and relies on the choice of basis. This parallels the concept of n-dimensional vector spaces, which are also isomorphic to R^n but lack a natural isomorphism. The example of homomorphisms from Z^2 to Z illustrates the non-canonical nature of these isomorphisms.

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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?


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The Attempt at a Solution

 
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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)
 

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