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.
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.
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PDFDOI: https://doi.org/10.2478/tatra.v59i0.335