Subgroups of finite Abelian groups having rank two via Goursat’s lemma

Abstract

Using Goursat’s lemma for groups, a simple representation and the invariant factor decompositions
of the subgroups of the group $\mathbb{Z}_m \times  \mathbb{Z}_n$ are deduced, where $m$ and $n$ are arbitrary
positive integers. As consequences, explicit formulas for the total number of subgroups,
the number of subgroups with a given invariant factor decomposition, and the number of
subgroups of a given order are obtained.