Uniquness and existence of free abelian group and construction

Question

Let $I$ be a set. A pair $(G,\epsilon)$ consisting of a abelian group $G$ and a map $\epsilon:I \to G$ is called a free abelian group over $I$ if and only if for all abelian groups $H$ and maps $\phi : I \to H$, there exists a unique homomorphism $\varphi: G \to H$, such that $\varphi\circ \epsilon = \phi$. Show for every set $I$ there exists a free abelian group over $I$ which is unique up to isomorphism.

I have a question about the construction of $G$.

My initial thought was to take finite linear combinations of elements of I and we define addition in the same way we add vectors over linear space and $\epsilon$ would simply be the inclusion map. Is there a reason why this construction does or does not work?

This does seem like the most obvious approach but the textbook I read actually defines $G$ to be the set of functions from $I \to Z$ which are finitely supported (each function takes the $0$ value for all but finitely many elements of $I$). This approach is definitely not intuitive or obvious at all.

What is the reason for this contruction instead?

Answer 1

Your idea, and the contruction from the textbook are pretty much the same thing. Suppose we have a function $f:I\to\mathbb{Z}$ with finite support. For each $i\in I$ let's write $f(i)=n_i$. Then we can denote $f$ as a formal sum:

$f=\sum\limits_{i\in I} n_i\cdot i$

And so we indeed get a formal "finite linear combination" of the elements of $I$ over $\mathbb{Z}$.

What's the problem with just defining $G$ as the set of such formal combinations? Well, many authors would just do it like this. But the question is, what is a formal linear combination? How do you formally define such a thing? If you think about it for a moment, it's not very clear how to give a mathematical definition for this object. On the other hand, a function $I\to\mathbb{Z}$ with compact support is indeed a mathematical object which we know how to define. And the formal linear combinations are just notations for such functions, as I described.

Same thing happens with polynomials. In most books a polynomial over a field $F$ (or over a ring, doesn't matter) is defined as a formal sum of the form $\sum\limits_{i=0}^n a_ix^i$. That's indeed how we think of polynomials. But again, this definition is not very formal. When I teach a course in algebra, I always define a polynomial as a function $\mathbb{N}\to F$ with compact support, and the "formal sums" are just a good notation.

Answer 2

If you look at the definition you quote and assume that $G$ is the group you constructed (i.e., inifinite linear combinations), and $H$ is the group with only finite combinations of elements from $I$ you will find it difficult (actually impossible) to construct hopmomorphism $\phi : G\rightarrow H$; indeed, how do you map an infinite combination to a finite one?

Another way to look at it: universal objects (which is the definition of $G$ in your quote) are usually minimal in certain respect. Using only finite combinations is enough to build a group containing $I$ with no dependencies between the elements except commutativity.