DOCUMENTA MATHEMATICA, Quadratic Forms LSU (2001), 165-181

Alexander Hahn

The Zassenhaus Decomposition for the Orthogonal Group: Properties and Applications

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 and used it to provide an alternative description of the spinor norm. This decomposition played a central role in the study of 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. See Hahn [6]. The current article develops the fundamental properties of the Zassenhaus decomposition, e.g., those of uniqueness and conjugacy, and applies them to sharpen and expand the analysis of [6].

2000 Mathematics Subject Classification: 20G15, 20G25, 20F05.

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