The Zassenhaus Decomposition for the Orthogonal Group: Properties and Applications

The orthogonal group of a quadratic space over a field has been the focus of intensive research over the past fifty years. As this article will indicate, it continues to be a source of interesting problems. Zassenhaus [17] constructed a decomposition for any element in the orthogonal group of a non-degenerate quadratic space over a field of characteristic not 2. This paper will develop the fundamental properties of this decomposition, e.g., those of uniqueness and conjugacy. It will also consider the question of the length of an element in the commutator subgroup of the orthogonal group with respect to the generating set of all elementary commutators of hyperplane reflections. The Zassenhaus decomposition plays a central role in the answer. Mathematics Subject Classification (2000): 20G15, 20G25, 20F05.