Editing Orthogonal group (section)

=== Dickson invariant ===
For orthogonal groups, the '''Dickson invariant''' is a homomorphism from the orthogonal group to the quotient group {{math|'''Z''' / 2'''Z'''}} (integers modulo 2), taking the value {{math|0}} in case the element is the product of an even number of reflections, and the value of 1 otherwise.<ref name=Knus224>{{citation | last=Knus | first=Max-Albert | title=Quadratic and Hermitian forms over rings | series=Grundlehren der Mathematischen Wissenschaften | volume=294 | location=Berlin etc. | publisher=[[Springer-Verlag]] | year=1991 | isbn=3-540-52117-8 | zbl=0756.11008 | page=224 }}</ref>

Algebraically, the Dickson invariant can be defined as {{math|1=''D''(''f'') = rank(''I'' − ''f'') modulo 2}}, where {{math|''I''}} is the identity {{harv|Taylor|1992|loc=Theorem 11.43}}. Over fields that are not of [[characteristic (algebra)|characteristic]] 2 it is equivalent to the determinant: the determinant is {{math|−1}} to the power of the Dickson invariant.
Over fields of characteristic 2, the determinant is always 1, so the Dickson invariant gives more information than the determinant.

The special orthogonal group is the [[Kernel (matrix)|kernel]] of the Dickson invariant<ref name=Knus224/> and usually has index 2 in {{math|O(''n'', ''F'' )}}.<ref>{{harv|Taylor|1992|loc=page 160}}</ref> When the characteristic of {{math|''F''}} is not 2, the Dickson Invariant is {{math|0}} whenever the determinant is {{math|1}}. Thus when the characteristic is not 2, {{math|SO(''n'', ''F'' )}} is commonly defined to be the elements of {{math|O(''n'', ''F'' )}} with determinant {{math|1}}. Each element in {{math|O(''n'', ''F'' )}} has determinant {{math|±1}}. Thus in characteristic 2, the determinant is always {{math|1}}.

The Dickson invariant can also be defined for [[Clifford group]]s and [[pin group]]s in a similar way (in all dimensions).