| dbp:proof
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- For studying the orthogonal group of , one can suppose that the matrix of the quadratic form is because, given a quadratic form, there is a basis where its matrix is diagonalizable. A matrix belongs to the orthogonal group if , that is, , , and . As and cannot be both zero , the second equation implies the existence of in , such that and . Reporting these values in the third equation, and using the first equation, one gets that , and thus the orthogonal group consists of the matrices
:
where and . Moreover, the determinant of the matrix is .
For further studying the orthogonal group, it is convenient to introduce a square root of . This square root belongs to if the orthogonal group is , and to otherwise. Setting , and , one has
:
If and are two matrices of determinant one in the orthogonal group then
:
This is an orthogonal matrix
with , and . Thus
:
It follows that the map is a homomorphism of the group of orthogonal matrices of determinant one into the multiplicative group of .
In the case of , the image is the multiplicative group of , which is a cyclic group of order .
In the case of , the above and are conjugate, and are therefore the image of each other by the Frobenius automorphism. This meant that and thus . For every such one can reconstruct a corresponding orthogonal matrix. It follows that the map is a group isomorphism from the orthogonal matrices of determinant 1 to the group of the -roots of unity. This group is a cyclic group of order which consists of the powers of , where is a primitive element of ,
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 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:
:
where is a primitive element of and is the multiplicative group of the element of norm one in ;
:
with and
In the real case, the corresponding isomorphisms are:
:
where is the circle of the complex numbers of norm one;
:
with and (en)
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