The Elements of the Orthogonal Group Ωn(V) as Products of Commutators of Symmetries Original Research Article

We will assume throughout thatFis a field of characteristic char F≠2 and thatVis a non-degenerate quadratic space overFof finite dimension dim V=n. The orthogonal group ofVis denotedOn(V) and Ωn(V) is its commutator subgroup. John Hsia, in reference to the classical fact that every element of Ωn(V) is a product of commutators of symmetries, asked the following very basic question: ForFa local field, does there exist a boundk, depending only onn, such that every element of Ωn(V) is a product ofksuch commutators or fewer? A corollary of the results of this article answers this question completely for a non-dyadicF: Every element in Ωn(V) is a product of [n/2] such commutators, except for a “handful” of elements which can be listed (all are certain types of involutions whenn≥6), where [n/2]+1 factors are required. (See Theorem 4.) Of course, Hsiaʹs question can be asked for anyF, and most of the analysis is carried out in the more general context.