  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.
