On the subgroups of finite Abelian groups of rank three

TL;DR

This paper characterizes all subgroups of finite Abelian groups of rank three, providing a formula for their total count and an asymptotic estimate for a specific case, advancing understanding of subgroup structures.

Contribution

It introduces a simple formula for counting subgroups of b2m b2n b2r in finite Abelian groups of rank three, including an asymptotic analysis.

Findings

Derived a formula for the total number of subgroups s(m,n,r).

Provided an asymptotic estimate for s(n,n,n).

Enhanced understanding of subgroup structures in rank three Abelian groups.

Abstract

We describe the subgroups of the group $\Z_m \times \Z_n \times \Z_r$ 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.