On subgroups of the special linear group containing the special orthogonal group (Q1062145)

Let V be an n-dimensional vector space over an arbitrary field K of characteristic not 2. Let \(O_ n\) be an orthogonal group of Witt index \(\geq 1\) and \(SL_ n\) be the special linear group over K. Let N be the normalizer of \(SL_ n\cap O_ n\) in \(SL_ n\). The main result is that N is a maximal subgroup of \(SL_ n\) for \(n\geq 3\). If \(n=2\) then a similar result holds provided the cardinality of K is at least 13.