 Solution

The structure theorem for finitely generated modules over a principal ideal domain applies to a Euclidean domain because every Euclidean domain is a principal ideal domain. In invariant-factor form, every finitely generated module $M$ has a decompositionwhere $r\ge 0$, and each $d_i$ is a nonzero nonunit. The rank of a module $r$ is unique and the invariant factors of a finitely generated module $d_i$ are unique up to multiplication by units. The zero module is allowed by taking $r=s=0$.

Every abelian group is a module over $\mathbb{Z}$, and a finite abelian group is finitely generated. Since $\mathbb{Z}$ is a Euclidean domain, the theorem applies. A nonzero free module summand would be infinite, so a finite group has $r=0$. Choosing positive integer representatives gives the finite abelian classificationThe ordered integers $d_i$ are uniquely determined. Splitting each $d_i$ into prime powers using the Chinese remainder theorem gives the equivalent elementary-divisor form as a direct sum of cyclic prime-power groups.