On the number of cyclic subgroups of a finite abelian group

TL;DR

This paper derives formulas for counting elements of fixed order and cyclic subgroups in finite abelian groups, highlighting multiplicative properties of related counting functions using number-theoretic methods.

Contribution

It provides simple number-theoretic formulas for counting elements and cyclic subgroups in finite abelian groups, clarifying their multiplicative properties.

Findings

Formulas for the number of elements of a fixed order

Formulas for the number of cyclic subgroups

Multiplicative properties of counting functions

Abstract

We prove by using simple number-theoretic arguments formulae concerning the number of elements of a fixed order and the number of cyclic subgroups of a direct product of several finite cyclic groups. We point out that certain multiplicative properties of related counting functions for finite Abelian groups are immediate consequences of these formulae.