A Universe of Sorts

If we have a finitely presented group, we can write any element as a product of the generators.. Say we have two genetors $g, h$ and some relations between them, we can have elements $gh$ , $ghgh$ , $gghh$ , $ghg^{-1}$ , and so on.
If the group is abelian, we can rearrange the strings to write them as $g^a h^b$ . For example, $ghgh = g^2h^2$ , and $ghg^{-1} = g^0h^1$ and so on.
Then, the only information about the element is carried by the powers of $g, h$ .
If $g$ has order $n$ and $h$ has order $m$ , then the powers live in $Z/nZ, Z/mZ$ .
Thus, the group above is isomorphic to $Z/nZ \times Z/mZ$ by rearranging and collecting powers.
The same argument works for any finitely generated abelian group.

§ Intuition for Why Finitely Presented Abelian Groups Are Isomorphic to Product of Cyclics