Unveiling Symmetries Patterns: A Study of Circular and Linear Harmonic Oscillator Chains

The purpose of this article is the study of the symmetries in a circular and linear harmonic oscillator chains system, and consequently use them as a means to find the eigenvalues of these configurations. Furthermore, a hidden ${\mathbb{Z}}_2$ group structure arises in both problems, showing how a degenerate spectrum in the circular case is attributable to the specific geometry producing a ${\mathbb{Z}}_N$ symmetry.