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arXiv · 1307.1414

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

Werner Georg Nowak·László Tóth
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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.

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Werner Georg Nowak, László Tóth. 2013-07-04. On the average number of subgroups of the group $\Z_m \times \Z_n$. https://doi.org/10.1142/s179304211350098x

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