Scherk's theorem on orthogonalities revisited (Q1072104)

The orthogonal group is generated by reflections. In 1950, P. Scherk determined for each isometry \(\pi\) the minimal number of reflections needed to express \(\pi\) as a product of reflections. He assumed that the characteristic of the field of scalars is distinct from 2. Several authors have obtained similar results for other classical groups and in the process reproved Scherk's theorem, including the case of characteristic 2. The author of the present paper gives a characteristic- free proof of Scherk's theorem followed by a detailed discussion of the case where the field of scalars is GF(2). This case has been neglected in most previous publications on the subject.