Editing Orthogonal group (section)

=== Characteristic different from two ===
Over a field of characteristic different from two, two [[quadratic form]]s are ''equivalent'' if their matrices are [[congruent matrices|congruent]], that is if a change of basis transforms the matrix of the first form into the matrix of the second form. Two equivalent quadratic forms have clearly the same orthogonal group.

The non-degenerate quadratic forms over a finite field of characteristic different from two are completely classified   into congruence classes, and it results from this classification that there is only one orthogonal group in odd dimension and two in even dimension.

More precisely, [[Witt's decomposition theorem]] asserts that (in characteristic different from two) every vector space equipped with a non-degenerate quadratic form {{mvar|Q}} can be decomposed as a direct sum of pairwise orthogonal subspaces
: <math>V = L_1 \oplus L_2 \oplus \cdots \oplus L_m \oplus W,</math>
where each {{mvar|L{{sub|i}}}} is a [[hyperbolic plane (quadratic forms)|hyperbolic plane]] (that is there is a basis such that the matrix of the restriction of {{mvar|Q}} to {{mvar|L{{sub|i}}}} has the form <math>\textstyle\begin{bmatrix}0&1\\1&0\end{bmatrix}</math>), and the restriction of {{mvar|Q}} to {{mvar|W}} is [[anisotropic quadratic form|anisotropic]] (that is, {{math|''Q''(''w'') ≠ 0}} for every nonzero {{mvar|w}} in {{mvar|W}}).

The [[Chevalley–Warning theorem]] asserts that, over a [[finite field]], the dimension of {{mvar|W}} is at most two.

If the dimension of {{mvar|V}} is odd, the dimension of {{mvar|W}} is thus equal to one, and its matrix is congruent either to <math>\textstyle\begin{bmatrix}1\end{bmatrix}</math> or to <math>\textstyle\begin{bmatrix}\varphi\end{bmatrix},</math> where {{mvar|{{varphi}}}} is a non-square scalar. It results that there is only one orthogonal group that is denoted {{math|O(2''n'' + 1, ''q'')}}, where {{mvar|q}} is the number of elements of the finite field (a power of an odd prime).<ref name=Wil6975>{{cite book | last=Wilson | first=Robert A. | title=The finite simple groups | zbl=1203.20012 | series=Graduate Texts in Mathematics | volume=251 | location=London | publisher=Springer | isbn=978-1-84800-987-5 | year=2009 | pages=69–75 }}</ref>

If the dimension of {{mvar|W}} is two and {{math|−1}} is not a square in the ground field (that is, if its number of elements {{mvar|q}} is congruent to 3 modulo 4), the matrix of the restriction of {{mvar|Q}} to {{mvar|W}} is congruent to either {{mvar|I}} or {{math|−''I''}}, where {{mvar|I}} is the 2×2 identity matrix. If the dimension of {{mvar|W}} is two and {{math|−1}} is a square in the ground field (that is, if {{mvar|q}} is congruent to 1, modulo 4) the matrix of the restriction of {{mvar|Q}} to {{mvar|W}} is congruent to <math>\textstyle\begin{bmatrix}1&0\\0&\varphi\end{bmatrix},</math> {{mvar|&phi;}} is any non-square scalar.

This implies that if the dimension of {{mvar|V}} is even, there are only two orthogonal groups, depending whether the dimension of {{mvar|W}} zero or two. They are denoted respectively {{math|O<sup>+</sup>(2''n'', ''q'')}} and {{math|O<sup>−</sup>(2''n'', ''q'')}}.<ref name=Wil6975 />

The orthogonal group {{math|O<sup>''&epsilon;''</sup>(2, ''q'')}} is a [[dihedral group]] of order {{math|2(''q''  − ''ε'')}}, where {{math|1=''ε'' = ±}}.
{{Math proof|drop=hidden|Title=Proof:|proof=
For studying the orthogonal group of {{math|O<sup>''ε''</sup>(2, ''q'')}}, one can suppose that the matrix of the quadratic form is <math>Q=\begin{bmatrix}1&0\\0&-\omega\end{bmatrix},</math> because, given a quadratic form, there is a basis where its matrix is diagonalizable. A matrix <math>A=\begin{bmatrix}a&b\\c&d\end{bmatrix}</math> belongs to the orthogonal group if {{math|1=''AQA''{{sup|T}} = Q}}, that is, {{math|1= ''a''{{sup|2}} − ''ωb''{{sup|2}} = 1}}, {{math|1=''ac'' − ''ωbd'' = 0}}, and {{math|1= ''c''{{sup|2}} − ''ωd''{{sup|2}} = −''ω''}}. As {{mvar|a}} and {{mvar|b}} cannot be both zero (because of the first equation), the second equation implies the existence of {{mvar|ε}} in {{math|'''F'''{{sub|''q''}}}}, such that {{math|1=''c'' = ''εωb''}} and {{math|1=''d'' = ''εa''}}. Reporting these values in the third equation, and using the first equation, one gets that {{math|1=''ε''{{sup|2}} = 1}}, and thus the orthogonal group consists of the matrices 
: <math>\begin{bmatrix}a&b\\\varepsilon\omega b&\varepsilon a\end{bmatrix},</math>
where {{math|1=''a''{{sup|2}} − ''ωb''{{sup|2}} = 1}} and {{math|1=''ε'' = ±1}}. Moreover, the determinant of the matrix is {{mvar|ε}}.

For further studying the orthogonal group, it is convenient to introduce a square root {{mvar|α}} of {{mvar|ω}}. This square root belongs to {{math|'''F'''{{sub|''q''}}}} if the orthogonal group is {{math|O<sup>+</sup>(2, ''q'')}}, and to {{math|'''F'''{{sub|''q''{{sup|2}}}}}} otherwise. Setting {{math|1=''x'' = ''a'' + ''αb''}}, and {{math|1=''y'' = ''a'' − ''αb''}}, one has
: <math>xy=1,\qquad a=\frac{x+y}2\qquad b=\frac{x-y}{2\alpha}.</math>

If <math>A_1=\begin{bmatrix}a_1&b_1\\\omega b_1&a_1\end{bmatrix}</math> and <math>A_2=\begin{bmatrix}a_2&b_2\\\omega b_2& a_2\end{bmatrix}</math> are two matrices of determinant one in the orthogonal group then 
: <math>A_1A_2=\begin{bmatrix}a_1a_2+\omega b_1b_2 & a_1b_2+ b_1 a_2\\
\omega b_1a_2 +\omega a_1b_2 & \omega b_1b_2+ a_1a_1\end{bmatrix}. </math>
This is an orthogonal matrix <math>\begin{bmatrix}a&b\\\omega b&a\end{bmatrix},</math>
with  {{math|1=''a'' = ''a''{{sub|1}}''a''{{sub|2}} + ''ωb''{{sub|1}}''b''{{sub|2}}}}, and {{math|1=''b'' = ''a''{{sub|1}}''b''{{sub|2}} + ''b''{{sub|1}}''a''{{sub|2}}}}. Thus 
: <math>a+\alpha b = (a_1+\alpha b_1)(a_2+\alpha b_2).</math>

It follows that the map {{math|(''a'', ''b'') ↦ ''a'' + ''αb''}} is a homomorphism of the group of orthogonal matrices  of determinant one into the multiplicative group of {{math|'''F'''{{sub|''q''{{sup|2}}}}}}.

In the case of {{math|O<sup>+</sup>(2''n'', ''q'')}}, the image is the multiplicative group of {{math|'''F'''{{sub|''q''}}}}, which is a cyclic group of order {{mvar|q}}.

In the case of {{math|O<sup>–</sup>(2''n'', ''q'')}}, the above {{mvar|x}} and {{mvar|y}} are [[Conjugate element (field theory)|conjugate]], and are therefore the image of each other by the [[Frobenius automorphism]]. This meant that <math>y=x^{-1}=x^q,</math> and thus {{math|1=''x''{{sup|''q''+1}} = 1}}. For every such {{mvar|x}} one can reconstruct a corresponding orthogonal matrix. It follows that the map <math>(a,b)\mapsto a+\alpha b</math> is a group isomorphism from the orthogonal matrices of determinant 1 to the group of the {{math|(''q'' + 1)}}-[[roots of unity]]. This group is a cyclic group of order {{math|''q'' + 1}} which consists of the powers of {{math|''g''{{sup|''q''−1}}}}, where {{mvar|g}} is a [[primitive element (finite field)|primitive element]] of {{math|'''F'''{{sub|''q''{{sup|2}}}}}},

For finishing the proof, it suffices to verify that the group all orthogonal matrices is not abelian, and is the semidirect product of the group {{math|{{mset|1, −1}}}} and the group of orthogonal matrices of determinant one.

The comparison of this proof with the real case may be illuminating.

Here two group isomorphisms are involved:
: <math>\begin{align}
\mathbf Z/(q+1)\mathbf Z &\to T\\
k&\mapsto g^{(q-1)k},
\end{align}</math>
where {{mvar|g}} is a primitive element of {{math|'''F'''{{sub|''q''{{sup|2}}}}}} and {{mvar|T}} is the multiplicative group of the element of norm one in {{math|'''F'''{{sub|''q''{{sup|2}}}}}} ;
: <math>\begin{align}
\mathbf T &\to \operatorname{SO}^+ (2, \mathbf F_q) \\
x&\mapsto \begin{bmatrix}a&b\\\omega b&a\end{bmatrix},
\end{align}</math>
with <math>a = \frac {x+x^{-1}}2</math> and <math>b = \frac {x- x^{-1}}{2\alpha}.</math>

In the real case, the corresponding isomorphisms are:
: <math>\begin{align}
\mathbf R/2\pi\mathbf R &\to C\\
\theta&\mapsto e^{i\theta},
\end{align}</math>
where {{mvar|C}} is the circle of the complex numbers of norm one;
: <math>\begin{align}
\mathbf C &\to \operatorname{SO}(2, \mathbf R) \\
x&\mapsto \begin{bmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{bmatrix},
\end{align}</math>
with <math>\cos\theta = \frac {e^{i\theta}+e^{-i\theta}}2</math> and <math>\sin\theta = \frac {e^{i\theta}-e^{-i\theta}}{2i}.</math>
}}

When the characteristic is not two, the order of the orthogonal groups are<ref>{{harv|Taylor|1992|p=141}}</ref>
: <math>\left|\operatorname{O}(2n + 1, q)\right| = 2q^{n^2}\prod_{i=1}^{n}\left(q^{2i} - 1\right),</math>
: <math>\left|\operatorname{O}^+(2n, q)\right| = 2q^{n(n-1)}\left(q^n-1\right)\prod_{i=1}^{n-1}\left(q^{2i} - 1\right),</math>
: <math>\left|\operatorname{O}^-(2n, q)\right| = 2q^{n(n-1)}\left(q^n+ 1\right)\prod_{i=1}^{n-1}\left(q^{2i} - 1\right).</math>
In characteristic two, the formulas are the same, except that the factor {{math|2}} of {{math|{{abs|O(2''n'' + 1, ''q'')}}}} must be removed.