On the subgroups of finite Abelian groups of rank three

Abstract

We describe the subgroups of the group ${\mathbb{Z}}_m\times {\mathbb{Z}}_n\times {\mathbb{Z}}_r$ and derive a simple formula for the total number $s(m,n,r)$ of the subgroups, where $m,n,r$ are arbitrary positive integers. An asymptotic formula for the function $n\mapsto s(n,n,n)$ is also deduced.

11 pages, 3 tables

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• Cyclicity degrees of finite groups