[Paper Review] On the number of cyclic subgroups of a finite abelian group

László Tóth|arXiv (Cornell University)|Mar 28, 2012

Limits and Structures in Graph Theory14 references19 citations

TL;DR

This paper provides a new number-theoretic proof for formulae counting the number of cyclic subgroups and elements of a given order in finite abelian groups, particularly in direct products of cyclic groups. It establishes that the number of cyclic subgroups of order δ in $ C_{n_1} imes imes C_{n_r} $ is given by $ c_\delta(n_1,\ldots,n_r) = \sum_{\text{lcm}(d_1,\ldots,d_r)=\delta} \phi(d_1)\cdots\phi(d_r) / \phi(\delta) $, and derives a multiplicative structure for these counting functions via Möbius inversion and Jordan 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.

Motivation & Objective

• To derive explicit formulae for the number of cyclic subgroups and elements of a fixed order in finite abelian groups using elementary number theory.
• To provide a direct, self-contained proof of Theorem 1, which gives a compact formula for the total number of cyclic subgroups in a direct product of cyclic groups.
• To establish the multiplicative structure of the counting functions for cyclic subgroups and elements via prime power decomposition and convolution properties.
• To connect the counting of cyclic subgroups to known arithmetic functions such as the Jordan function and von Sterneck function.

Proposed method

• Use of Möbius inversion to derive the number of elements of order δ in $ C_{n_1} \times \cdots \times C_{n_r} $, expressed as $ o_\delta(n_1,\ldots,n_r) = \sum_{e|\delta} \gcd(e,n_1)\cdots\gcd(e,n_r) \mu(\delta/e) $.
• Derivation of an alternative formula via divisor sums: $ o_\delta(n_1,\ldots,n_r) = \sum_{\text{lcm}(d_1,\ldots,d_r)=\delta} \phi(d_1)\cdots\phi(d_r) $, using the identity $ \sum_{d|n} \phi(d) = n $.
• Application of the formula $ c_\delta(n_1,\ldots,n_r) = o_\delta(n_1,\ldots,n_r)/\phi(\delta) $ to compute the number of cyclic subgroups of order δ.
• Proof of multiplicativity of the counting function $ c(n_1,\ldots,n_r) $ via convolution of multiplicative arithmetic functions and prime power decomposition.
• Derivation of a special case for p-groups: $ c_{p^\nu}(p^{a_1},\ldots,p^{a_r}) = \frac{1}{p^{\nu-1}(p-1)} \left( p^{\sum \min(\nu,a_i)} - p^{\sum \min(\nu-1,a_i)} \right) $.
• Identification of the Jordan function $ \phi_r(\delta) $ as the number of elements of order δ in $ C_n^r $, linking group-theoretic counts to arithmetic functions.

Experimental results

Research questions

• RQ1 How can the number of cyclic subgroups of a given order in a finite abelian group be computed using elementary number-theoretic methods?
• RQ2 What is the precise formula for the number of elements of order δ in a direct product of cyclic groups $ C_{n_1} \times \cdots \times C_{n_r} $?
• RQ3 How does the multiplicative structure of the counting function for cyclic subgroups arise from the group's primary decomposition?
• RQ4 What is the relationship between the number of cyclic subgroups and classical arithmetic functions such as Euler’s totient and Jordan functions?
• RQ5 Can the formula for the number of cyclic subgroups be derived directly from the element count via Möbius inversion and divisor sum identities?

Key findings

• The number of elements of order δ in $ C_{n_1} \times \cdots \times C_{n_r} $ is given by $ \sum_{e|\delta} \gcd(e,n_1)\cdots\gcd(e,n_r) \mu(\delta/e) $, which is a new derivation using Möbius inversion.
• An equivalent formula is $ \sum_{\text{lcm}(d_1,\ldots,d_r)=\delta} \phi(d_1)\cdots\phi(d_r) $, showing a direct link between cyclic subgroup counts and least common multiples of divisors.
• The number of cyclic subgroups of order δ is $ c_\delta(n_1,\ldots,n_r) = \frac{1}{\phi(\delta)} \sum_{\text{lcm}(d_1,\ldots,d_r)=\delta} \phi(d_1)\cdots\phi(d_r) $, which generalizes known results for p-groups.
• For $ C_n^r $, the number of elements of order δ is the Jordan function $ \phi_r(\delta) = \delta^r \prod_{p|\delta} (1 - p^{-r}) $, providing a group-theoretic interpretation of this arithmetic function.
• The total number of cyclic subgroups in $ C_{n_1} \times \cdots \times C_{n_r} $ is multiplicative over prime powers, and given by $ c(n_1,\ldots,n_r) = \sum_{\delta|n} c_\delta(n_1,\ldots,n_r) $, with $ n = \text{lcm}(n_1,\ldots,n_r) $.
• The formula $ c(n_1,\ldots,n_r) = \sum_{d_1|n_1,\ldots,d_r|n_r} \frac{\phi(d_1)\cdots\phi(d_r)}{\phi(\text{lcm}(d_1,\ldots,d_r))} $ is re-derived via elementary means, confirming its validity through combinatorial number theory.