Finite orthogonal groups and periodicity of links

For a prime number $q\mathrm{\ne }2$ and $r>0$ we study, whether there exists an isometry of order $q^r$ acting on a free ${\mathbb{Z}}_{p^k}$-module equipped with a scalar product. We investigate, whether there exists such an isometry with no non-zero fixed points. Both questions are completely answered in this paper if $p\mathrm{\ne }2,q$. As an application we refine Naik’s criterion for periodicity of links in $S^3$. The periodicity criterion we obtain is effectively computable and gives concrete restrictions for periodicity of low-crossing knots.