On the average number of subgroups of the group ${\mathbb{Z}}_m\times {\mathbb{Z}}_n$

Abstract

Let ${\mathbb{Z}}_m$ 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 ${\mathbb{Z}}_m\times {\mathbb{Z}}_n$ 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 ${\mathbb{Z}}_m\times {\mathbb{Z}}_n$ having rank two.

14 pages

Cited by in corpus (1)

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