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

Abstract

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

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