2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/115610The problem of computing the index of a coincidence isometry of the hyper cubic lattice $\mathbb{Z}^{n}$ is considered. The normal form of a rational orthogonal matrix is analyzed in detail, and explicit formulas for the index of certain coincidence isometries of $\mathbb{Z}^{n}$ are obtained. These formulas generalize the known results for $n\le 4$.AMS-Latex, preprint 10 pagesGroup TheoryMetric Geometry20H15; 51F15text