Commutators of symmetries in characteristic 2
Let \(V\) be a vector space of dimension \(n\) over a field \(K\) where \(|K|>2\) equipped with a quadratic form \(Q\). Then \(Q(v+w) = Q(v)+Q(w) + f(v,w)\) for some bilinear form \(f\). The space \(V\) is called nonsingular if \(V\cap V^\perp = 0\) where the orthogonal complement \(\perp\) is with respect to \(f\). The main result of the paper asserts the following length theorem: Let \(V\) be a nonsingular vector space over a field \(K\) of characteristic \(2\) with \(|K|>3\). Suppose \(K\) is perfect and \(\pi\in SO(V)\) (the special orthogonal group) with \(\dim B(\pi) =2d\), where the path of \(\pi\) is \(B(\pi): =\text{im} (\pi-1)\). Then the length of \(\pi\) with respect to the symmetry commutators is \(d\) if \(B(\pi)\) is not totally isotropic; otherwise it is \(d+1\). The proof requires an analysis different from the case of characteristic \(2\).
scientific article; zbMATH DE number 5860486
scientific article