Structure theorem for finitely generated abelian groups

Question

How can we use fundamental theorem of finitely generated abelian groups to list all abelian groups of order 16 up to isomorphism.

Answer 1

This is a minor complement to Arturo's answer.

The number of abelian groups of order $p_1^{n_1}\cdots p_k^{n_k}$ (where the $p_i$ are distinct primes and the $n_i$ are positive integers), considered up to isomorphism, is $$ P(n_1)\cdots P(n_k), $$ where $P(n)$ is the number of partitions of $n$.

As explained by Arturo, the Structure Theorem tells us which group corresponds to a given partition (and a given prime).

To wit: If the prime is $p$ and the parts of the partition are $n_1,\dots,n_k$, then the group is the product of the cyclic groups of order $p^{n_i}$.

(I'm using implicitly the fact that a finite abelian group can we written in a unique way as a product of $p$-groups.)

Answer 2

The Theorem tells you that every finitely generated abelian group can be written as a direct sum of cyclic groups which, under certain conditions, is unique. So find all possible direct sums of cyclic groups that satisfy those conditions and give you groups of order $16$ to get the answer.

(There are two versions of the structure theorem, but the conditions coincide for groups of prime power order).