Single conjugacy classes of isometries in orthogonal groups over local fields

Abstract

All isometries $\sigma$ in a quadratic space over a non-archimedean local field of characteristic not 2 satisfying that any isometry $\tau$ which is conjugate to $\sigma$ in the general linear group is conjugate to $\sigma$ in the orthogonal group are determined. This extends \cite[Theorem 2.1]{Mil} to arbitary cases.