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.

Key words and phrases. Abelian group of rank three, subgroup, number of subgroups, multiplicative arithmetic function, asymptotic formula.