Let be a finite abelian group with for each .

- Find a presentation for .
- Prove that if is a group containing commuting elements such that for each , then there is a unique group homomorphism such that .

- We claim that , where . Indeed, the “standard basis” elements , consisting of in the -th coordinate and 1 in all other coordinates, satisfy these relations.
- Note that every element of can be written uniquely as . Define , where the product on the right hand side is as computed inside . It is clear that for each . Moreover, is a homomorphism because the commute pairwise.
Finally, suppose is a homomorphism such that for each . Then . Thus (products computed inside ), so that is unique.