§ Intuition for why finitely presented abelian groups are isomorphic to product of cyclics

• 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.