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Using Goursat's lemma for groups, a simple representation and the invariantfactor decompositions of the subgroups of the group Z_m x Z_n are deduced,where m and n are arbitrary positive integers. As consequences, explicitformulas for the total number of subgroups, the number of subgroups with agiven invariant factor decomposition, and the number of subgroups of a givenorder are obtained.

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