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.