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Orthogonal group

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Inmathematics, the orthogonal group in dimension n, denoted O(n), is the groupofdistance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing transformations. The orthogonal group is sometimes called the general orthogonal group, by analogy with the general linear group. Equivalently, it is the group of n × n orthogonal matrices, where the group operation is given by matrix multiplication (an orthogonal matrix is a real matrix whose inverse equals its transpose). The orthogonal group is an algebraic group and a Lie group. It is compact.

The orthogonal group in dimension n has two connected components. The one that contains the identity element is a normal subgroup, called the special orthogonal group, and denoted SO(n). It consists of all orthogonal matrices of determinant 1. This group is also called the rotation group, generalizing the fact that in dimensions 2 and 3, its elements are the usual rotations around a point (in dimension 2) or a line (in dimension 3). In low dimension, these groups have been widely studied, see SO(2), SO(3) and SO(4). The other component consists of all orthogonal matrices of determinant −1. This component does not form a group, as the product of any two of its elements is of determinant 1, and therefore not an element of the component.

By extension, for any field F, an n × n matrix with entries in F such that its inverse equals its transpose is called an orthogonal matrix over F. The n × n orthogonal matrices form a subgroup, denoted O(n, F), of the general linear group GL(n, F); that is $${\displaystyle \operatorname {O} (n,F)=\left\{Q\in \operatorname {GL} (n,F)\mid Q^{\mathsf {T}}Q=QQ^{\mathsf {T}}=I\right\}.}$$

More generally, given a non-degenerate symmetric bilinear formorquadratic form^[1] on a vector space over a field, the orthogonal group of the form is the group of invertible linear maps that preserve the form. The preceding orthogonal groups are the special case where, on some basis, the bilinear form is the dot product, or, equivalently, the quadratic form is the sum of the square of the coordinates.

All orthogonal groups are algebraic groups, since the condition of preserving a form can be expressed as an equality of matrices.

Name[edit]

The name of "orthogonal group" originates from the following characterization of its elements. Given a Euclidean vector space E of dimension n, the elements of the orthogonal group O(n) are, up toauniform scaling (homothecy), the linear maps from EtoE that map orthogonal vectors to orthogonal vectors.

In Euclidean geometry[edit]

The orthogonal O(n) is the subgroup of the general linear group GL(n, R), consisting of all endomorphisms that preserve the Euclidean norm; that is, endomorphisms g such that ${\displaystyle \|g($x)\|=\|x\|.}

Let E(n) be the group of the Euclidean isometries of a Euclidean space S of dimension n. This group does not depend on the choice of a particular space, since all Euclidean spaces of the same dimension are isomorphic. The stabilizer subgroup of a point x ∈ S is the subgroup of the elements g ∈ E(n) such that g(x) = x. This stabilizer is (or, more exactly, is isomorphic to) O(n), since the choice of a point as an origin induces an isomorphism between the Euclidean space and its associated Euclidean vector space.

There is a natural group homomorphism p from E(n)toO(n), which is defined by

${\displaystyle p($g)(y-x)=g(y)-g(x),}

where, as usual, the subtraction of two points denotes the translation vector that maps the second point to the first one. This is a well defined homomorphism, since a straightforward verification shows that, if two pairs of points have the same difference, the same is true for their images by g (for details, see Affine space § Subtraction and Weyl's axioms).

The kernelofp is the vector space of the translations. So, the translations form a normal subgroupofE(n), the stabilizers of two points are conjugate under the action of the translations, and all stabilizers are isomorphic to O(n).

Moreover, the Euclidean group is a semidirect productofO(n) and the group of translations. It follows that the study of the Euclidean group is essentially reduced to the study of O(n).

Special orthogonal group[edit]

By choosing an orthonormal basis of a Euclidean vector space, the orthogonal group can be identified with the group (under matrix multiplication) of orthogonal matrices, which are the matrices such that

${\displaystyle QQ^{\mathsf {T}}=I.}$

It follows from this equation that the square of the determinantofQ equals 1, and thus the determinant of Q is either 1or−1. The orthogonal matrices with determinant 1 form a subgroup called the special orthogonal group, denoted SO(n), consisting of all direct isometriesofO(n), which are those that preserve the orientation of the space.

SO(n) is a normal subgroup of O(n), as being the kernel of the determinant, which is a group homomorphism whose image is the multiplicative group {−1, +1}. This implies that the orthogonal group is an internal semidirect productofSO(n) and any subgroup formed with the identity and a reflection.

The group with two elements {±I} (where I is the identity matrix) is a normal subgroup and even a characteristic subgroupofO(n), and, if n is even, also of SO(n). If n is odd, O(n) is the internal direct productofSO(n) and {±I}.

The group SO(2)isabelian (whereas SO(n) is not abelian when n >2). Its finite subgroups are the cyclic group C_kofk-fold rotations, for every positive integer k. All these groups are normal subgroups of O(2) and SO(2).

Canonical form[edit]

For any element of O(n) there is an orthogonal basis, where its matrix has the form

${\displaystyle {\begin{bmatrix}{\begin{matrix}R_{1}&&\\&\ddots &\\&&R_{k}\end{matrix}}&0\\0&{\begin{matrix}\pm 1&&\\&\ddots &\\&&\pm 1\end{matrix}}\\\end{bmatrix}},}$

where the matrices R₁, ..., R_k are 2-by-2 rotation matrices, that is matrices of the form

${\displaystyle {\begin{bmatrix}a&b\\-b&a\end{bmatrix}},}$

with a² + b² = 1.

This results from the spectral theorem by regrouping eigenvalues that are complex conjugate, and taking into account that the absolute values of the eigenvalues of an orthogonal matrix are all equal to 1.

The element belongs to SO(n) if and only if there are an even number of −1 on the diagonal. A pair of eigenvalues −1 can be identified with a rotation by π and a pair of eigenvalues +1 can be identified with a rotation by 0.

The special case of n = 3 is known as Euler's rotation theorem, which asserts that every (non-identity) element of SO(3) is a rotation about a unique axis–angle pair.

Reflections[edit]

Reflections are the elements of O(n) whose canonical form is

${\displaystyle {\begin{bmatrix}-1&0\\0&I\end{bmatrix}},}$

where I is the (n − 1) × (n − 1) identity matrix, and the zeros denote row or column zero matrices. In other words, a reflection is a transformation that transforms the space in its mirror image with respect to a hyperplane.

In dimension two, every rotation can be decomposed into a product of two reflections. More precisely, a rotation of angle θ is the product of two reflections whose axes form an angle of θ / 2.

A product of up to n elementary reflections always suffices to generate any element of O(n). This results immediately from the above canonical form and the case of dimension two.

The Cartan–Dieudonné theorem is the generalization of this result to the orthogonal group of a nondegenerate quadratic form over a field of characteristic different from two.

The reflection through the origin (the map v ↦ −v) is an example of an element of O(n) that is not a product of fewer than n reflections.

Symmetry group of spheres[edit]

The orthogonal group O(n) is the symmetry group of the (n − 1)-sphere (for n = 3, this is just the sphere) and all objects with spherical symmetry, if the origin is chosen at the center.

The symmetry group of a circleisO(2). The orientation-preserving subgroup SO(2) is isomorphic (as a real Lie group) to the circle group, also known as U(1), the multiplicative group of the complex numbers of absolute value equal to one. This isomorphism sends the complex number exp(φ i) = cos(φ) + i sin(φ)ofabsolute value 1 to the special orthogonal matrix

${\displaystyle {\begin{bmatrix}\cos(\varphi )&-\sin(\varphi )\\\sin(\varphi )&\cos(\varphi )\end{bmatrix}}.}$

In higher dimension, O(n) has a more complicated structure (in particular, it is no longer commutative). The topological structures of the n-sphere and O(n) are strongly correlated, and this correlation is widely used for studying both topological spaces.

Group structure[edit]

The groups O(n) and SO(n) are real compact Lie groupsofdimension n(n − 1) / 2. The group O(n) has two connected components, with SO(n) being the identity component, that is, the connected component containing the identity matrix.

As algebraic groups[edit]

The orthogonal group O(n) can be identified with the group of the matrices A such that A^TA = I. Since both members of this equation are symmetric matrices, this provides n(n + 1) / 2 equations that the entries of an orthogonal matrix must satisfy, and which are not all satisfied by the entries of any non-orthogonal matrix.

This proves that O(n) is an algebraic set. Moreover, it can be proved^{[citation needed]} that its dimension is

${\displaystyle {\frac {n(n-1)}{2}}=n^{2}-{\frac {n(n+1)}{2}},}$

which implies that O(n) is a complete intersection. This implies that all its irreducible components have the same dimension, and that it has no embedded component. In fact, O(n) has two irreducible components, that are distinguished by the sign of the determinant (that is det(A) = 1ordet(A) = −1). Both are nonsingular algebraic varieties of the same dimension n(n − 1) / 2. The component with det(A) = 1isSO(n).

Maximal tori and Weyl groups[edit]

Amaximal torus in a compact Lie group G is a maximal subgroup among those that are isomorphic to T^k for some k, where T = SO(2) is the standard one-dimensional torus.^[2]

InO(2n) and SO(2n), for every maximal torus, there is a basis on which the torus consists of the block-diagonal matrices of the form

${\displaystyle {\begin{bmatrix}R_{1}&&0\\&\ddots &\\0&&R_{n}\end{bmatrix}},}$

where each R_j belongs to SO(2). In O(2n + 1) and SO(2n + 1), the maximal tori have the same form, bordered by a row and a column of zeros, and 1 on the diagonal.

The Weyl groupofSO(2n + 1) is the semidirect product ${\displaystyle \{\pm 1\}^{n}\rtimes S_{n}}$ of a normal elementary abelian 2-subgroup and a symmetric group, where the nontrivial element of each {±1} factor of {±1}ⁿ acts on the corresponding circle factor of T × {1} by inversion, and the symmetric group S_n acts on both {±1}ⁿ and T × {1} by permuting factors. The elements of the Weyl group are represented by matrices in O(2n) × {±1}. The S_n factor is represented by block permutation matrices with 2-by-2 blocks, and a final 1 on the diagonal. The {±1}ⁿ component is represented by block-diagonal matrices with 2-by-2 blocks either

${\displaystyle {\begin{bmatrix}1&0\\0&1\end{bmatrix}}\quad {\text{or}}\quad {\begin{bmatrix}0&1\\1&0\end{bmatrix}},}$

with the last component ±1 chosen to make the determinant 1.

The Weyl group of SO(2n) is the subgroup ${\displaystyle H_{n-1}\rtimes S_{n}<\{\pm 1\}^{n}\rtimes S_{n}}$ of that of SO(2n + 1), where H_n−1 < {±1}ⁿ is the kernel of the product homomorphism {±1}ⁿ → {±1} given by ${\displaystyle \left(\varepsilon _{1},\ldots ,\varepsilon _{n}\right)\mapsto \varepsilon _{1}\cdots \varepsilon _{n}}$; that is, H_n−1 < {±1}ⁿ is the subgroup with an even number of minus signs. The Weyl group of SO(2n) is represented in SO(2n) by the preimages under the standard injection SO(2n) → SO(2n + 1) of the representatives for the Weyl group of SO(2n + 1). Those matrices with an odd number of ${\displaystyle {\begin{bmatrix}0&1\\1&0\end{bmatrix}}}$ blocks have no remaining final −1 coordinate to make their determinants positive, and hence cannot be represented in SO(2n).

Topology[edit]

This section may be confusing or unclear to readers. In particular, most notations are undefined; no context for explaining why these consideration belong to the article. Moreover, the section consists essentially in a list of advanced results without providing the information that is needed for a non-specialist for verifying them (no reference, no link to articles about the methods of computation that are used, no sketch of proofs. Please help clarify the section. There might be a discussion about this on the talk page. (November 2019) (Learn how and when to remove this message)

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Low-dimensional topology[edit]

The low-dimensional (real) orthogonal groups are familiar spaces:

• O(1) = S⁰, a two-point discrete space
• SO(1) = {1}
• SO(2)isS¹
• SO(3)isRP^{3 [3]}
• SO(4)isdoubly coveredbySU(2) × SU(2) = S³ × S³.

Fundamental group[edit]

In terms of algebraic topology, for n >2 the fundamental groupofSO(n, R)iscyclic of order 2,^[4] and the spin group Spin(n) is its universal cover. For n = 2 the fundamental group is infinite cyclic and the universal cover corresponds to the real line (the group Spin(2) is the unique connected 2-fold cover).

Homotopy groups[edit]

Generally, the homotopy groups π_k(O) of the real orthogonal group are related to homotopy groups of spheres, and thus are in general hard to compute. However, one can compute the homotopy groups of the stable orthogonal group (aka the infinite orthogonal group), defined as the direct limit of the sequence of inclusions:

${\displaystyle \operatorname {O} (0)\subset \operatorname {O} ($1)\subset \operatorname {O} (2)\subset \cdots \subset O=\bigcup _{k=0}^{\infty }\operatorname {O} (k)}

Since the inclusions are all closed, hence cofibrations, this can also be interpreted as a union. On the other hand, Sⁿ is a homogeneous space for O(n + 1), and one has the following fiber bundle:

${\displaystyle \operatorname {O} ($n)\to \operatorname {O} (n+1)\to S^{n},}

which can be understood as "The orthogonal group O(n + 1) acts transitively on the unit sphere Sⁿ, and the stabilizer of a point (thought of as a unit vector) is the orthogonal group of the perpendicular complement, which is an orthogonal group one dimension lower." Thus the natural inclusion O(n) → O(n + 1)is(n − 1)-connected, so the homotopy groups stabilize, and π_k(O(n + 1)) = π_k(O(n)) for n > k + 1: thus the homotopy groups of the stable space equal the lower homotopy groups of the unstable spaces.

From Bott periodicity we obtain Ω⁸O ≅ O, therefore the homotopy groups of O are 8-fold periodic, meaning π_{k + 8}(O) = π_k(O), and one need only to list the lower 8 homotopy groups:

${\displaystyle {\begin{aligned}\pi _{0}($O)&=\mathbf {Z} /2\mathbf {Z} \\\pi _{1}(O)&=\mathbf {Z} /2\mathbf {Z} \\\pi _{2}(O)&=0\\\pi _{3}(O)&=\mathbf {Z} \\\pi _{4}(O)&=0\\\pi _{5}(O)&=0\\\pi _{6}(O)&=0\\\pi _{7}(O)&=\mathbf {Z} \end{aligned}}}

Relation to KO-theory[edit]

Via the clutching construction, homotopy groups of the stable space O are identified with stable vector bundles on spheres (up to isomorphism), with a dimension shift of 1: π_k(O) = π_{k + 1}(BO). Setting KO = BO × Z = Ω⁻¹O × Z (to make π₀ fit into the periodicity), one obtains:

${\displaystyle {\begin{aligned}\pi _{0}($KO)&=\mathbf {Z} \\\pi _{1}(KO)&=\mathbf {Z} /2\mathbf {Z} \\\pi _{2}(KO)&=\mathbf {Z} /2\mathbf {Z} \\\pi _{3}(KO)&=0\\\pi _{4}(KO)&=\mathbf {Z} \\\pi _{5}(KO)&=0\\\pi _{6}(KO)&=0\\\pi _{7}(KO)&=0\end{aligned}}}

Computation and interpretation of homotopy groups[edit]

Low-dimensional groups[edit]

The first few homotopy groups can be calculated by using the concrete descriptions of low-dimensional groups.

• π₀(O) = π₀(O(1)) = Z / 2Z, from orientation-preserving/reversing (this class survives to O(2) and hence stably)
• π₁(O) = π₁(SO(3)) = Z / 2Z, which is spin comes from SO(3) = RP³ = S³ / (Z / 2Z).
• π₂(O) = π₂(SO(3)) = 0, which surjects onto π₂(SO(4)); this latter thus vanishes.

Lie groups[edit]

From general facts about Lie groups, π₂(G) always vanishes, and π₃(G) is free (free abelian).

Vector bundles[edit]

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π₀(KO) is a vector bundle over S⁰, which consists of two points. Thus over each point, the bundle is trivial, and the non-triviality of the bundle is the difference between the dimensions of the vector spaces over the two points, so π₀(KO) = Z is the dimension.

Loop spaces[edit]

Using concrete descriptions of the loop spaces in Bott periodicity, one can interpret the higher homotopies of O in terms of simpler-to-analyze homotopies of lower order. Using π₀, O and O/U have two components, KO = BO × Z and KSp = BSp × Z have countably many components, and the rest are connected.

Interpretation of homotopy groups[edit]

In a nutshell:^[5]

• π₀(KO) = Z is about dimension
• π₁(KO) = Z / 2Z is about orientation
• π₂(KO) = Z / 2Z is about spin
• π₄(KO) = Z is about topological quantum field theory.

Let R be any of the four division algebras R, C, H, O, and let L_R be the tautological line bundle over the projective line RP¹, and [L_R] its class in K-theory. Noting that RP¹ = S¹, CP¹ = S², HP¹ = S⁴, OP¹ = S⁸, these yield vector bundles over the corresponding spheres, and

• π₁(KO) is generated by [L_R]
• π₂(KO) is generated by [L_C]
• π₄(KO) is generated by [L_H]
• π₈(KO) is generated by [L_O]

From the point of view of symplectic geometry, π₀(KO) ≅ π₈(KO) = Z can be interpreted as the Maslov index, thinking of it as the fundamental group π₁(U/O) of the stable Lagrangian GrassmannianasU/O ≅ Ω⁷(KO), so π₁(U/O) = π₁₊₇(KO).

Whitehead tower[edit]

The orthogonal group anchors a Whitehead tower:

${\displaystyle \cdots \rightarrow \operatorname {Fivebrane} ($n)\rightarrow \operatorname {String} (n)\rightarrow \operatorname {Spin} (n)\rightarrow \operatorname {SO} (n)\rightarrow \operatorname {O} (n)}

which is obtained by successively removing (killing) homotopy groups of increasing order. This is done by constructing short exact sequences starting with an Eilenberg–MacLane space for the homotopy group to be removed. The first few entries in the tower are the spin group and the string group, and are preceded by the fivebrane group. The homotopy groups that are killed are in turn π₀(O) to obtain SO from O, π₁(O) to obtain Spin from SO, π₃(O) to obtain String from Spin, and then π₇(O) and so on to obtain the higher order branes.

Of indefinite quadratic form over the reals[edit]

Over the real numbers, nondegenerate quadratic forms are classified by Sylvester's law of inertia, which asserts that, on a vector space of dimension n, such a form can be written as the difference of a sum of p squares and a sum of q squares, with p + q = n. In other words, there is a basis on which the matrix of the quadratic form is a diagonal matrix, with p entries equal to 1, and q entries equal to −1. The pair (p, q) called the inertia, is an invariant of the quadratic form, in the sense that it does not depend on the way of computing the diagonal matrix.

The orthogonal group of a quadratic form depends only on the inertia, and is thus generally denoted O(p, q). Moreover, as a quadratic form and its opposite have the same orthogonal group, one has O(p, q) = O(q, p).

The standard orthogonal group is O(n) = O(n, 0) = O(0, n). So, in the remainder of this section, it is supposed that neither p nor q is zero.

The subgroup of the matrices of determinant 1 in O(p, q) is denoted SO(p, q). The group O(p, q) has four connected components, depending on whether an element preserves orientation on either of the two maximal subspaces where the quadratic form is positive definite or negative definite. The component of the identity, whose elements preserve orientation on both subspaces, is denoted SO⁺(p, q).

The group O(3, 1) is the Lorentz group that is fundamental in relativity theory. Here the 3 corresponds to space coordinates, and 1 corresponds to the time coordinate.

Of complex quadratic forms[edit]

Over the field Cofcomplex numbers, every non-degenerate quadratic forminn variables is equivalent to x₁² + ... + x_n². Thus, up to isomorphism, there is only one non-degenerate complex quadratic space of dimension n, and one associated orthogonal group, usually denoted O(n, C). It is the group of complex orthogonal matrices, complex matrices whose product with their transpose is the identity matrix.

As in the real case, O(n, C) has two connected components. The component of the identity consists of all matrices of determinant 1inO(n, C); it is denoted SO(n, C).

The groups O(n, C) and SO(n, C) are complex Lie groups of dimension n(n − 1) / 2 over C (the dimension over R is twice that). For n ≥ 2, these groups are noncompact. As in the real case, SO(n, C) is not simply connected: For n >2, the fundamental groupofSO(n, C)iscyclic of order 2, whereas the fundamental group of SO(2, C)isZ.

Over finite fields[edit]

Characteristic different from two[edit]

Over a field of characteristic different from two, two quadratic forms are equivalent if their matrices are 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 Q can be decomposed as a direct sum of pairwise orthogonal subspaces

${\displaystyle V=L_{1}\oplus L_{2}\oplus \cdots \oplus L_{m}\oplus W,}$

where each L_i is a hyperbolic plane (that is there is a basis such that the matrix of the restriction of QtoL_i has the form ${\displaystyle \textstyle {\begin{bmatrix}0&1\\1&0\end{bmatrix}}}$), and the restriction of QtoWisanisotropic (that is, Q(w) ≠ 0 for every nonzero winW).

The Chevalley–Warning theorem asserts that, over a finite field, the dimension of W is at most two.

If the dimension of V is odd, the dimension of W is thus equal to one, and its matrix is congruent either to ${\displaystyle \textstyle {\begin{bmatrix}1\end{bmatrix}}}$ or to ${\displaystyle \textstyle {\begin{bmatrix}\varphi \end{bmatrix}},}$ where 𝜑 is a non-square scalar. It results that there is only one orthogonal group that is denoted O(2n + 1, q), where q is the number of elements of the finite field (a power of an odd prime).^[6]

If the dimension of W is two and −1 is not a square in the ground field (that is, if its number of elements q is congruent to 3 modulo 4), the matrix of the restriction of QtoW is congruent to either Ior–I, where I is the 2×2 identity matrix. If the dimension of W is two and −1 is a square in the ground field (that is, if q is congruent to 1, modulo 4) the matrix of the restriction of QtoW is congruent to ${\displaystyle \textstyle {\begin{bmatrix}1&0\\0&\varphi \end{bmatrix}},}$ φ is any non-square scalar.

This implies that if the dimension of V is even, there are only two orthogonal groups, depending whether the dimension of W zero or two. They are denoted respectively O⁺(2n, q) and O⁻(2n, q).^[6]

The orthogonal group O^ε(2, q) is a dihedral group of order 2(q − ε), where ε = ±.

${\displaystyle {\begin{bmatrix}a&b\\\varepsilon \omega b&\varepsilon a\end{bmatrix}},}$

where a² – ωb² = 1 and ε = ±1. Moreover, the determinant of the matrix is ε.

For further studying the orthogonal group, it is convenient to introduce a square root αofω. This square root belongs to F_q if the orthogonal group is O⁺(2, q), and to F_q² otherwise. Setting x = a + αb, and y = a – αb, one has

${\displaystyle xy=1,\qquad a={\frac {x+y}{2}}\qquad b={\frac {x-y}{2\alpha }}.}$

If${\displaystyle A_{1}={\begin{bmatrix}a_{1}&b_{1}\\\omega b_{1}&a_{1}\end{bmatrix}}}$ and ${\displaystyle A_{2}={\begin{bmatrix}a_{2}&b_{2}\\\omega b_{2}&a_{2}\end{bmatrix}}}$ are two matrices of determinant one in the orthogonal group then

${\displaystyle A_{1}A_{2}={\begin{bmatrix}a_{1}a_{2}+\omega b_{1}b_{2}&a_{1}b_{2}+b_{1}a_{2}\\\omega b_{1}a_{2}+\omega a_{1}b_{2}&\omega b_{1}b_{2}+a_{1}a_{1}\end{bmatrix}}.}$

This is an orthogonal matrix ${\displaystyle {\begin{bmatrix}a&b\\\omega b&a\end{bmatrix}},}$ with a = a₁a₂ + ωb₁b₂, and b = a₁b₂ + b₁a₂. Thus

${\displaystyle a+\alpha b=(a_{1}+\alpha b_{1})(a_{2}+\alpha b_{2}).}$

It follows that the map (a, b) ↦ a + αb is a homomorphism of the group of orthogonal matrices of determinant one into the multiplicative group of F_q².

In the case of O⁺(2n, q), the image is the multiplicative group of F_q, which is a cyclic group of order q.

In the case of O^–(2n, q), the above x and y are conjugate, and are therefore the image of each other by the Frobenius automorphism. This meant that ${\displaystyle y=x^{-1}=x^{q},}$ and thus x^q+1 = 1. For every such x one can reconstruct a corresponding orthogonal matrix. It follows that the map ${\displaystyle (a,b)\mapsto a+\alpha b}$ is a group isomorphism from the orthogonal matrices of determinant 1 to the group of the (q + 1)-roots of unity. This group is a cyclic group of order q + 1 which consists of the powers of g^q−1, where g is a primitive elementofF_q²,

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 {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:

${\displaystyle {\begin{aligned}\mathbf {Z} /(q+1)\mathbf {Z} &\to T\\k&\mapsto g^{(q-1)k},\end{aligned}}}$

where g is a primitive element of F_q² and T is the multiplicative group of the element of norm one in F_q² ;

${\displaystyle {\begin{aligned}\mathbf {T} &\to \operatorname {SO} ^{+}(2,\mathbf {F} _{q})\\x&\mapsto {\begin{bmatrix}a&b\\\omega b&a\end{bmatrix}},\end{aligned}}}$

with ${\displaystyle a={\frac {x+x^{-1}}{2}}}$ and ${\displaystyle b={\frac {x-x^{-1}}{2\alpha }}.}$

In the real case, the corresponding isomorphisms are:

${\displaystyle {\begin{aligned}\mathbf {R} /2\pi \mathbf {R} &\to C\\\theta &\mapsto e^{i\theta },\end{aligned}}}$

where C is the circle of the complex numbers of norm one;

${\displaystyle {\begin{aligned}\mathbf {C} &\to \operatorname {SO} (2,\mathbf {R} )\\x&\mapsto {\begin{bmatrix}\cos \theta &\sin \theta \\-\sin \theta &\cos \theta \end{bmatrix}},\end{aligned}}}$

with ${\displaystyle \cos \theta ={\frac {e^{i\theta }+e^{-i\theta }}{2}}}$ and ${\displaystyle \sin \theta ={\frac {e^{i\theta }-e^{-i\theta }}{2i}}.}$