Representing and counting the subgroups of the group Z_m x Z_n

We deduce a simple representation and the invariant factor decompositions of the subgroups of the group ${\mathbb{Z}}_m\times {\mathbb{Z}}_n$, where $m$ and $n$ are arbitrary positive integers. We obtain formulas for the total number of subgroups and the number of subgroups of a given order.