§ 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 and some relations between them, we can have elements , , , , and so on.
- If the group is abelian, we can rearrange the strings to write them as . For example, , and and so on.
- Then, the only information about the element is carried by the powers of .
- If has order and has order , then the powers live in .
- Thus, the group above is isomorphic to by rearranging and collecting powers.
- The same argument works for any finitely generated abelian group.