The number of subgroups of the group $\Bbb{Z}_m\times \Bbb{Z}_n \times \Bbb{Z}_r \times \Bbb{Z}_s$

TL;DR

This paper derives explicit formulas for counting subgroups and subgroups of specific orders in the direct product of four cyclic groups, using group and number theory, and proposes two conjectures.

Contribution

It provides new explicit formulas for subgroup enumeration in a four-fold cyclic group product, advancing understanding in group theory.

Findings

Formulas for total number of subgroups

Formulas for number of subgroups of given order

Two conjectures related to subgroup counts

Abstract

We deduce direct formulas for the total number of subgroups and the number of subgroups of a given order of the group $\Bbb{Z}_m\times \Bbb{Z}_n \times \Bbb{Z}_r \times \Bbb{Z}_s$, where $m,n,r,s\in \Bbb{N}$. The proofs are by some simple group theoretical and number theoretical arguments based on Goursat's lemma for groups. Two conjectures are also formulated.