The Fundamental Theorem of Finitely Generated Abelian Groups states:
Let $G$ be a finitely generated abelian group. Then it decomposes as follows: $$ G \cong Z^r \times Z_{n_1} \times Z_{n_2} \times \cdot \cdot \cdot \times Z_{n_s}, $$ for some integers $r,n_1,n_2, \cdot \cdot \cdot,n_s$ satisfying the following conditions:
$r≥0$ and $n_i≥2$ for all $i$, and
$n_{i+1}|n_i$ for $1≤i≤s−1$.
The decomposition of $G$ satisfying these conditions is unique.
If $r=0$, I know what these groups will look like. They are finite, and elements in the group are tuples where the $i$th element is an element of the $ith$ cylic group.
But it $r>0$, then the group is infinite. This means that there are some indices of the tuple where we know that the element will have one of a finite option of values. But, for $r$ elements of the tuple, we don't know anything about the element, because there are infinite choices.
I was wondering if this was a good way to think about these groups? Do we treat elements in these infinite groups like $(s+r) -$ tuples?