On the average number of subgroups of the group $\Z_m \times \Z_n$

TL;DR

This paper derives asymptotic formulas for the total and cyclic subgroups of groups formed by direct products of residue class groups, focusing on their average counts as m and n grow large.

Contribution

It provides new asymptotic formulas for sums of subgroup counts in direct product groups of residue classes, including cases with gcd greater than one.

Findings

Asymptotic formulas for sums of subgroup counts up to x

Results include cases with gcd(m,n)>1

Focus on groups with rank two

Abstract

Let $\Z_m$ be the group of residue classes modulo $m$. Let $s(m,n)$ and $c(m,n)$ denote the total number of subgroups of the group $\Z_m \times \Z_n$ and the number of its cyclic subgroups, respectively, where $m$ and $n$ are arbitrary positive integers. We derive asymptotic formulas for the sums $\sum_{m,n\le x} s(m,n)$, $\sum_{m,n\le x} c(m,n)$ and for the corresponding sums restricted to $\gcd(m,n)>1$, i.e., concerning the groups $\Z_m \times \Z_n$ having rank two.