 Solution

The Fundamental theorem of finitely generated abelian groups says that every finitely generated abelian group is isomorphic toThe rank of an abelian group $r$ and the invariant factors $d_j$ are uniquely determined, with the finite summand allowed to be absent. Equivalently, one can decompose the torsion subgroup into cyclic groups of prime-power orders. The infinite cyclic part and torsion part must both be included in the statement.