[Paper Review] Representing and counting the subgroups of the group Z_m x Z_n

Mario Hampejs, Nicki Holighaus+2 more|arXiv (Cornell University)|Nov 8, 2012

Advanced Algebra and Geometry21 references4 citations

TL;DR

This paper provides a complete characterization of subgroups of the finite abelian group ℤₘ × ℤₙ using a simple representation and invariant factor decompositions. It derives explicit, multiplicative formulas for the total number of subgroups and the number of subgroups of a given order, generalizing known results for p-groups and enabling efficient computation via arithmetic functions and Dirichlet convolutions.

ABSTRACT

We deduce a simple representation and the invariant factor decompositions of the subgroups of the group $\Bbb{Z}_m imes \Bbb{Z}_n$, where $m$ and $n$ are arbitrary positive integers. We obtain formulas for the total number of subgroups and the number of subgroups of a given order.

Motivation & Objective

• To provide a complete and simple representation of all subgroups of ℤₘ × ℤₙ for arbitrary positive integers m and n.
• To determine the invariant factor decompositions of these subgroups, enabling structural classification.
• To derive explicit, multiplicative formulas for the total number of subgroups s(m,n) and the number of subgroups of a given order sₖ(m,n).
• To generalize known results for p-groups to arbitrary ℤₘ × ℤₙ via prime factorization and multiplicative number-theoretic techniques.
• To support applications in time-frequency analysis by enabling efficient enumeration and selection of subgroups with desired properties.

Proposed method

• Uses a parametrization of subgroups via triples (a,b,t) with divisibility and congruence conditions to represent all subgroups of ℤₘ × ℤₙ.
• Applies the Chinese Remainder Theorem and prime factorization to reduce the problem to p-groups ℤₚᵃ × ℤₚᵇ, leveraging multiplicativity.
• Employs number-theoretic tools such as the Möbius function μ, Euler’s totient function φ, and the Dedekind psi function ψ to derive closed-form expressions.
• Utilizes Dirichlet convolution and the Busche-Ramanujan identity to transform sums over divisors into compact arithmetic functions.
• Applies Goursat’s lemma and matrix-based methods (e.g., Hermite normal form) in special cases to validate subgroup representations.
• Derives formulas for cyclic subgroups and subgroups of a given exponent using gcd and lcm conditions on parameters.

Experimental results

Research questions

• RQ1 How can all subgroups of ℤₘ × ℤₙ be systematically represented and classified using elementary group and number theory?
• RQ2 What is the exact number of subgroups of ℤₘ × ℤₙ, and how does it depend on the prime factorizations of m and n?
• RQ3 What is the number of subgroups of a given order k in ℤₘ × ℤₙ, and how can this be computed efficiently?
• RQ4 How do the subgroup structures of ℤₘ × ℤₙ relate to applications in time-frequency analysis and lattice sampling?
• RQ5 Can the number of cyclic subgroups and subgroups of a given exponent in ℤₙ × ℤₙ be expressed in terms of multiplicative arithmetic functions?

Key findings

• The total number of subgroups s(m,n) is multiplicative and given by s(m,n) = ∏ᵣ s(pⱼᵃʲ, pⱼᵇʲ), where u = gcd(m,n) and v = lcm(m,n) have prime factorizations u = ∏ pⱼᵃʲ, v = ∏ pⱼᵇʲ.
• For a p-group ℤₚᵃ × ℤₚᵇ with 0 ≤ a ≤ b, the number of subgroups is s(pᵃ,pᵇ) = [(b−a+1)pᵃ⁺² − (b−a−1)pᵃ⁺¹ − (a+b+3)p + (a+b+1)] / (p−1)².
• The number of subgroups of order pᶜ in ℤₚᵃ × ℤₚᵇ is given by a piecewise formula depending on the relative size of c with respect to a and b, involving the sum (pᶜ⁺¹−1)/(p−1) or similar expressions.
• The number of cyclic subgroups of ℤₘ × ℤₙ is c(m,n) = ∑_{lcm(d,e)=gcd(m,n)} gcd(d,e), which simplifies to c(m,n) = ∑_{ℓk|gcd(m,n)} ℓ·2^ω(k) via substitution and Möbius inversion.
• The number of subgroups of exponent δ in ℤₙ × ℤₙ is E₆(n) = c(δ), matching the number of cyclic subgroups of order δ in ℤₙ × ℤₙ.
• The number of subgroups of index n in ℤ×ℤ is σ(n), the sum of divisors of n, confirming a known result via the derived parametrization.