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

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