Editing Orthogonal group (section)

=== Orthogonal groups of characteristic 2 ===
Over fields of characteristic 2 orthogonal groups often exhibit special behaviors, some of which are listed in this section. (Formerly these groups were known as the '''hypoabelian groups''', but this term is no longer used.)
* Any orthogonal group over any field is generated by reflections, except for a unique example where the vector space is 4-dimensional over the field with 2 elements and the [[Witt index]] is 2.<ref>{{harv|Grove|2002|loc=Theorem 6.6 and 14.16}}</ref> A reflection in characteristic two has a slightly different definition.  In characteristic two, the reflection orthogonal to a vector {{math|'''u'''}} takes a vector {{math|'''v'''}} to {{math|'''v''' + ''B''('''v''', '''u''')/''Q''('''u''') · '''u'''}} where {{math|''B''}} is the bilinear form{{clarify|reason=This seems like it is the associated polar form B′(x,y)=Q(x+y)−Q(x)−Q(y) (the name 'associated bilinear form' is used variously to mean B′ or B=B′/2). When expressed in the same terms (e.g. in terms of Q), the expression for a reflection is the same for all cases.|date=May 2020}} and {{math|''Q''}} is the quadratic form associated to the orthogonal geometry.  Compare this to the [[Householder reflection]] of odd characteristic or characteristic zero, which takes {{math|'''v'''}} to {{math|'''v''' − 2·''B''('''v''', '''u''')/''Q''('''u''') · '''u'''}}.
* The [[center of a group|center]] of the orthogonal group usually has order 1 in characteristic 2, rather than 2, since {{math|1=''I'' = −''I''}}.
* In odd dimensions {{math|2''n'' + 1}} in characteristic 2, orthogonal groups over [[perfect field]]s are the same as [[symplectic group]]s in dimension {{math|2''n''}}. In fact the symmetric form is alternating in characteristic 2, and as the dimension is odd it must have a kernel of dimension 1, and the quotient by this kernel is a symplectic space of dimension {{math|2''n''}}, acted upon by the orthogonal group.
* In even dimensions in characteristic 2 the orthogonal group is a subgroup of the symplectic group, because the symmetric bilinear form of the quadratic form is also an alternating form.