Modular Grids and Lattices of Cyclic Subgroups: Arithmetic Fibers Associated with GCD and LCM

Abstract

This work introduces a finite modular-grid framework for encoding pairs and finite families of subgroups of a cyclic group. Let Cm=⟨R⟩ be the cyclic group of order m. To each point of (Z/mZ)2, we associate cyclic subgroups determined by the corresponding residue classes. Their generated subgroup and intersection are described through the greatest common divisor (GCD) and least common multiple (LCM) of the associated divisors of m. A canonical GCD–LCM map from the modular grid to the divisor lattice is introduced, and exact formulas for the cardinalities of its fibers are obtained. These formulas admit a prime-local factorization expressed in terms of Euler’s totient function. The construction is extended to (Z/mZ)k, where the local fiber problem is interpreted through the minimum and maximum of truncated p-adic valuations. The framework provides a finite connection between modular grids, cyclic subgroup lattices, divisor lattices, and arithmetic enumeration.