Products of reflections in the kernel of the spinorial norm (Q750611)

Let V be a finite-dimensional vector space over a finite field F of characteristic \(\neq 2\), f a symmetric bilinear form on V, O the corresponding orthogonal group and \(O'\) the kernel of the spinorial norm. As is well known, each element of O can be written as a product of reflections, and the minimal number of reflections needed in such a representation has been determined. The present paper treats this problem for \(O'\). It is shown that \(O'\) is generated by reflections; if \(| F| =3\), the extra assumption dim \(V\geq 6\) is needed, a counterexample with dim V\(=3\) being given. Further, if \(| F| \neq 3\), for each \(A\in O'\) the minimal number of reflections in \(O'\) needed to represent A is determined, both for non-degenerate f and for degenerate f.