Orthogonal group

In mathematics, the orthogonal group of dimension n, denoted O(n), is the group of distance-preserving transformations of a Euclidean space of dimension n that preserve a fixed point, where the group operation is given by composing transformations. Equivalently, it is the group of n×n orthogonal matrices, where the group operation is given by matrix multiplication, and an orthogonal matrix is a real matrix whose inverse equals its transpose.

The determinant of an orthogonal matrix being either 1 or −1, an important subgroup of O(n) is the special orthogonal group, denoted SO(n), of the orthogonal matrices of determinant 1. This group is also called the rotation group, because, 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 term "orthogonal group" may also refer to a generalization of the above case: the group of invertible linear operators that preserve a non-degenerate symmetric bilinear form or quadratic form[1] on a vector space over a field. In particular, when the bilinear form is the scalar product on the vector space F n of dimension n over a field F, with quadratic form the sum of squares, then the corresponding orthogonal group, denoted O(n, F ), is the set of n × n orthogonal matrices with entries from F, with the group operation of matrix multiplication. This is a subgroup of the general linear group GL(n, F ) given by

\( \mathrm{O}(n,F) = \{ Q \in \mathrm{GL}(n, F) \mid Q^\mathsf{T} Q = Q Q^\mathsf{T} = I \}

where QT is the transpose of Q and I is the identity matrix.

This article mainly discusses the orthogonal groups of quadratic forms that may be expressed over some bases as the dot product; over the reals, they are the positive definite quadratic forms. Over the reals, for any non-degenerate quadratic form, there is a basis, on which the matrix of the form is a diagonal matrix such that the diagonal entries are either 1 or −1. Thus the orthogonal group depends only on the numbers of 1 and of −1, and is denoted O(p, q), where p is the number of ones and q the number of negative ones. For details, see indefinite orthogonal group.

The derived subgroup Ω(n, F ) of O(n, F) is an often studied object because, when F is a finite field, Ω(n, F ) is often[clarification needed] a central extension of a finite simple group.

Both O(n, F ) and SO(n, F ) are algebraic groups, because the condition that a matrix be orthogonal, i.e. have its own transpose as inverse, can be expressed as a set of polynomial equations in the entries of the matrix. The Cartan–Dieudonné theorem describes the structure of the orthogonal group for a non-singular form.

Name

The determinant of any orthogonal matrix is either 1 or −1. The orthogonal n-by-n matrices with determinant 1 form a normal subgroup of O(n, F ) known as the special orthogonal group SO(n, F ), consisting of all proper rotations. (More precisely, SO(n, F ) is the kernel of the Dickson invariant, discussed below.). By analogy with GL–SL (general linear group, special linear group), the orthogonal group is sometimes called the general orthogonal group and denoted GO, though this term is also sometimes used for indefinite orthogonal groups O(p, q). The term rotation group can be used to describe either the special or general orthogonal group.
Even and odd dimension

The structure of the orthogonal group differs in certain aspects between even and odd dimensions – for example, over ordered fields (such as R) the −I element is orientation-preserving in even dimensions, but orientation-reversing in odd dimensions. When this distinction wishes to be emphasized, the groups are generally denoted O(2k) and O(2k + 1), reserving n for the dimension of the space (n = 2k or n = 2k + 1). The letters p or r are also used, indicating the rank of the corresponding Lie algebra; in odd dimension the corresponding Lie algebra is \mathfrak{so}(2r + 1), while in even dimension the Lie algebra is \( \mathfrak{so}(2r). \)
Difference between O(n) and SO(n) in even dimensions

In two dimensions O(2) is all such rotations about the origin and all reflections along a line through the origin. while SO(2) is the group of all rotations about the origin.

These groups are closely related: not only is SO(2) a subgroup of O(2) because any two reflections gives a rotation.

More generally an even number of reflections gives a rotation, and in n-dimensions all rotations can be generated this way.

To get a "reflection through the origin" you can reflect along each of the axes. Two axes in two dimensions, so two reflections which is a rotation, and the same is true in any dimension. The 'reflection through the origin' is not a reflection in the usual sense in even dimensions, it's a rotation. It's usually a particularly interesting rotation: in 2D it's the only rotation which when done twice gives the identity, so is its own inverse - this is true in higher dimensions though other rotations have the same property. In 4D it's isoclinic, and if that classification were generalised it would be isoclinic in higher dimensions too.
Over the real number field

Over the field R of real numbers, the orthogonal group O(n, R) and the special orthogonal group SO(n, R) are often simply denoted by O(n) and SO(n) if no confusion is possible. They form real compact Lie groups of dimension n(n − 1)/2. O(n, R) has two connected components, with SO(n, R) being the identity component, i.e., the connected component containing the identity matrix.
Geometric interpretation

The real orthogonal and real special orthogonal groups have the following geometric interpretations:

O(n, R) is a subgroup of the Euclidean group E(n), the group of isometries of Rn; it contains those that leave the origin fixed – O(n, R) = E(n) ∩ GL(n, R). It is the symmetry group of the sphere (n = 3) or (n − 1)-sphere and all objects with spherical symmetry, if the origin is chosen at the center.

SO(n, R) is a subgroup of E+(n), which consists of direct isometries, i.e., isometries preserving orientation; it contains those that leave the origin fixed – SO(n, R) = E+(n) ∩ GL(n, R) = E(n) ∩ GL+(n, R). It is the rotation group of the sphere and all objects with spherical symmetry, if the origin is chosen at the center.

{±I} is a normal subgroup and even a characteristic subgroup of O(n, R), and, if n is even, also of SO(n, R). If n is odd, O(n, R) is the internal direct product of SO(n, R) and {±I}. For every positive integer k the cyclic group Ck of k-fold rotations is a normal subgroup of O(2, R) and SO(2, R).

Relative to suitable orthogonal bases, the isometries are of the form:

\( \begin{bmatrix} \begin{matrix}R_1 & & \\ & \ddots & \\ & & R_k\end{matrix} & 0 \\ 0 & \begin{matrix}\pm 1 & & \\ & \ddots & \\ & & \pm 1\end{matrix} \\ \end{bmatrix} \)

where the matrices R1, ..., Rk are 2-by-2 rotation matrices in orthogonal planes of rotation. As a special case, known as Euler's rotation theorem, any (non-identity) element of SO(3, R) is rotation about a uniquely defined axis.

The orthogonal group is generated by reflections (two reflections give a rotation), as in a Coxeter group,[note 1] and elements have length at most n (require at most n reflections to generate; this follows from the above classification, noting that a rotation is generated by 2 reflections, and is true more generally for indefinite orthogonal groups, by the Cartan–Dieudonné theorem). A longest element (element needing the most reflections) is reflection through the origin (the map v ↦ −v), though so are other maximal combinations of rotations (and a reflection, in odd dimension).

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

\( \begin{bmatrix}\cos(\phi)&-\sin(\phi)\\ \sin(\phi)&\cos(\phi)\end{bmatrix}. \)

The group SO(3, R), understood as the set of rotations of 3-dimensional space, is of major importance in the sciences and engineering, and there are numerous charts on SO(3).
Maximal tori and Weyl groups

A maximal torus T for SO(2n), of rank n, is given by the block-diagonal matrices

\( \begin{bmatrix} R_1 & & 0\\ & \ddots & \\ 0& & R_n \end{bmatrix}, \)

where the Rj are 2-by-2 rotation matrices. The image T ×{1} of the same torus under the block-diagonal inclusion

\( \mathrm{SO}(2n) \cong \mathrm{SO}(2n) \times \{1\} < \mathrm{SO}(2n+1) \)

is a maximal torus for SO(2n+1). The Weyl group of SO(2n+1) is the semidirect product \{\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 Sn 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 Sn 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

\( \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix} \quad \text{or} \quad \begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix} , \)

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

The Weyl group of SO(2n) is the subgroup H_{n-1} \rtimes S_n < \{\pm 1\}^n \rtimes S_n of that of SO(2n + 1), where Hn−1 < {±1}n is the kernel of the product homomorphism {±1}n → {±1} given by (\epsilon_1,\ldots,\epsilon_n) \mapsto \epsilon_1 \cdots \epsilon_n; that is Hn−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 \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).
Low-dimensional topology

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

O(1) = S0, a two-point discrete space
SO(1) = {1}
SO(2) is S1
SO(3) is RP3
SO(4) is double covered by SU(2) × SU(2) = S3 × S3.

Homotopy groups

In terms of algebraic topology, for n > 2 the fundamental group of SO(n, R) is cyclic of order 2, 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 2-fold cover).

Generally, the homotopy groups πk(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:

\( \mathrm{O}(0) \subset \mathrm{O}(1)\subset \mathrm{O}(2)\subset\cdots\subset O = \bigcup_{k=0}^\infty \mathrm{O}(k) \)

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:

\( \mathrm{O}(n) \to \mathrm{O}(n+1) \to S^n \)

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 πk(O(n+1)) = πk(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 πk + 8(O) = πk(O), and one needs only to compute the lower 8 homotopy groups:

\( \begin{align} \pi_0 (O) &= \mathbf Z/2\\ \pi_1 (O) &= \mathbf Z/2\\ \pi_2 (O) &= 0\\ \pi_3 (O) &= \mathbf Z\\ \pi_4 (O) &= 0\\ \pi_5 (O) &= 0\\ \pi_6 (O) &= 0\\ \pi_7 (O) &= \mathbf Z \end{align} \)

Relation to KO-theory

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: πk(O) = πk + 1(BO). Setting KO = BO × Z = Ω−1O × Z (to make π0 fit into the periodicity), one obtains:

\( \begin{align} \pi_0 (KO) &= \mathbf Z\\ \pi_1 (KO) &= \mathbf Z/2\\ \pi_2 (KO) &= \mathbf Z/2\\ \pi_3 (KO) &= 0\\ \pi_4 (KO) &= \mathbf Z\\ \pi_5 (KO) &= 0\\ \pi_6 (KO) &= 0\\ \pi_7 (KO) &= 0 \) \end{align}

Computation and interpretation of homotopy groups
Low-dimensional groups

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

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

Lie groups

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

From the vector bundle point of view, π0(KO) is vector bundles over S0, which is 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 dimension.
Loop spaces

Using concrete descriptions of the loop spaces in Bott periodicity, one can interpret higher homotopy of O as lower homotopy of simple to analyze spaces. 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

In a nutshell:[2]

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

Let R be any of four division rings R, C, H, O, and let LR be the tautological line bundle over the projective line RP1, and [LR] 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 [LR]
π2(KO) is generated by [LC]
π4(KO) is generated by [LH]
π8(KO) is generated by [LO]

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 Grassmannian as U/O ≅ Ω7(KO), so π1(U/O) = π1 + 7(KO).
Over the complex number field

Over the field C of complex numbers, O(n, C) and SO(n, C) are complex Lie groups of dimension n(n − 1)/2 over C (it means the dimension over R is twice that). O(n, C) has two connected components, and SO(n, C) is the connected component containing the identity matrix. For n ≥ 2 these groups are noncompact.

Just as in the real case SO(n, C) is not simply connected. For n > 2 the fundamental group of SO(n, C) is cyclic of order 2 whereas the fundamental group of SO(2, C) is infinite cyclic.
Over finite fields

Orthogonal groups can also be defined over finite fields Fq, where is a power of a prime p.

Over finite fields of characteristic not equal to 2, orthogonal groups come in two types in even dimension: O+(2n, q) and O−(2n, q); and one type in odd dimension: O(2n + 1, q).[3]

If V is the vector space on which the orthogonal group G acts, it can be written as a direct orthogonal sum as follows:

\( V = L_1 \oplus L_2 \oplus \cdots \oplus L_m \oplus W, \)

where Li are hyperbolic lines and W contains no singular vectors. If W is the zero subspace, then G is of plus type. If W is one-dimensional then G has odd dimension. If W has dimension 2, G is of minus type.

In the special case where n = 1, Oϵ(2, q) is a dihedral group of order 2(q − ϵ).

We have the following formulas for the order of O(n, q), when the characteristic is not two:

\( |\mathrm{O}(2n+1,q)|=2q^n\prod_{i=0}^{n-1}(q^{2n}-q^{2i}). \)

If −1 is a square in Fq

\( |\mathrm{O}(2n,q)|=2(q^n-1)\prod_{i=1}^{n-1}(q^{2n}-q^{2i}). \)

If −1 is a non-square in Fq

\( |\mathrm{O}(2n,q)|=2(q^n+(-1)^{n+1})\prod_{i=1}^{n-1}(q^{2n}-q^{2i}). \)

The Dickson invariant

For orthogonal groups, the Dickson invariant is a homomorphism from the orthogonal group to the quotient group Z/2Z (integers modulo 2), taking the value 0 in case the element is the product of an even number of reflections, and the value of 1 otherwise.[4]

Algebraically, the Dickson invariant can be defined as D(f) = rank(I − f) modulo 2, where I is the identity (Taylor 1992, Theorem 11.43). Over fields that are not of characteristic 2 it is equivalent to the determinant: the determinant is −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 of the Dickson invariant[4] and usually has index 2 in O(n, F ).[5] When the characteristic of F is not 2, the Dickson Invariant is 0 whenever the determinant is 1. Thus when the characteristic is not 2, SO(n, F ) is commonly defined to be the elements of O(n, F ) with determinant 1. Each element in O(n, F ) has determinant ±1. Thus in characteristic 2, the determinant is always 1.

The Dickson invariant can also be defined for Clifford groups and Pin groups in a similar way (in all dimensions).
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.[6] Note that a reflection in characteristic two has a slightly different definition. In characteristic two, the reflection orthogonal to a vector u takes a vector v to v + B(v, u)/Q(u) · u where B is the bilinear form and Q is the quadratic form associated to the orthogonal geometry. Compare this to the Householder reflection of odd characteristic or characteristic zero, which takes v to v − 2·B(v, u)/Q(u) · u.

The center of the orthogonal group usually has order 1 in characteristic 2, rather than 2, since I = −I.

In odd dimensions 2n + 1 in characteristic 2, orthogonal groups over perfect fields are the same as symplectic groups in dimension 2n. 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 2n, 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.

The spinor norm

The spinor norm is a homomorphism from an orthogonal group over a field F to the quotient group F ∗/F∗ 2 (the multiplicative group of the field F up to square elements), that takes reflection in a vector of norm n to the image of n in F∗/F∗ 2.[7]

For the usual orthogonal group over the reals it is trivial, but it is often non-trivial over other fields, or for the orthogonal group of a quadratic form over the reals that is not positive definite.
Galois cohomology and orthogonal groups

In the theory of Galois cohomology of algebraic groups, some further points of view are introduced. They have explanatory value, in particular in relation with the theory of quadratic forms; but were for the most part post hoc, as far as the discovery of the phenomena is concerned. The first point is that quadratic forms over a field can be identified as a Galois H1, or twisted forms (torsors) of an orthogonal group. As an algebraic group, an orthogonal group is in general neither connected nor simply-connected; the latter point brings in the spin phenomena, while the former is related to the discriminant.

The 'spin' name of the spinor norm can be explained by a connection to the spin group (more accurately a pin group). This may now be explained quickly by Galois cohomology (which however postdates the introduction of the term by more direct use of Clifford algebras). The spin covering of the orthogonal group provides a short exact sequence of algebraic groups.

\( 1 \rightarrow \mu_2 \rightarrow \mathrm{Pin}_V \rightarrow \mathrm{O_V} \rightarrow 1 \)

Here μ2 is the algebraic group of square roots of 1; over a field of characteristic not 2 it is roughly the same as a two-element group with trivial Galois action. The connecting homomorphism from H0(OV), which is simply the group OV(F) of F-valued points, to H1(μ2) is essentially the spinor norm, because H1(μ2) is isomorphic to the multiplicative group of the field modulo squares.

There is also the connecting homomorphism from H1 of the orthogonal group, to the H2 of the kernel of the spin covering. The cohomology is non-abelian, so that this is as far as we can go, at least with the conventional definitions.
Lie algebra

The Lie algebra corresponding to Lie groups O(n, F ) and SO(n, F ) consists of the skew-symmetric n × n matrices, with the Lie bracket [ , ] given by the commutator. One Lie algebra corresponds to both groups. It is often denoted by \mathfrak{o}(n, F) or \mathfrak{so}(n, F), and called the orthogonal Lie algebra or special orthogonal Lie algebra. Over real numbers, these Lie algebras for different n are the compact real forms of two of the four families of semisimple Lie algebras: in odd dimension Bk, where n = 2k + 1, while in even dimension Dr, where n = 2r.

More intrinsically, given a vector space with an inner product, the special orthogonal Lie algebra is given by the bivectors on the space, which are sums of simple bivectors (2-blades) v ∧ w. The correspondence is given by the map \( \mathbf{v} \wedge \mathbf{w} \mapsto \mathbf{v}^* \otimes \mathbf{w} - \mathbf{w}^* \otimes \mathbf{v} \), where v∗ is the covector dual to the vector v; in coordinates these are exactly the elementary skew-symmetric matrices.

Over real numbers, this characterization is used in interpreting the curl of a vector field (naturally a 2-vector) as an infinitesimal rotation or "curl", hence the name. Generalizing the inner product with a nondegenerate form yields the indefinite orthogonal Lie algebras \mathfrak{so}(p, q).

The representation theory of the orthogonal Lie algebras includes both representations corresponding to linear representations of the orthogonal groups, and representations corresponding to projective representations of the orthogonal groups (linear representations of spin groups), the so-called spin representation, which are important in physics.

Related groups

The orthogonal groups and special orthogonal groups have a number of important subgroups, supergroups, quotient groups, and covering groups. These are listed below.

The inclusions O(n) ⊂ U(n) ⊂ Sp(n) = USp(2n) and USp(n) ⊂ U(n) ⊂ O(2n) are part of a sequence of 8 inclusions used in a geometric proof of the Bott periodicity theorem, and the corresponding quotient spaces are symmetric spaces of independent interest – for example, U(n)/O(n) is the Lagrangian Grassmannian.
Lie subgroups

In physics, particularly in the areas of Kaluza–Klein compactification, it is important to find out the subgroups of the orthogonal group. The main ones are:

\( \mathrm{O}(n) \supset \mathrm{O}(n-1) \) – preserve an axis
\( \mathrm{O}(2n) \supset \mathrm{U}(n) \supset \mathrm{SU}(n) \) – U(n) are those that preserve a compatible complex structure or a compatible symplectic structure – see 2-out-of-3 property; SU(n) also preserves a complex orientation.
\( \mathrm{O}(2n) \supset \mathrm{USp}(n) \)
\( \mathrm{O}(7) \supset \mathrm{G}_2 \)

Lie supergroups

The orthogonal group O(n) is also an important subgroup of various Lie groups:

\( \mathrm{U}(n) \supset \mathrm{SU}(n) \supset \mathrm{O}(n) \)
\( \mathrm{USp}(2n) \supset \mathrm{O}(n) \)
\(\mathrm{G}_2 \supset \mathrm{O}(3) \)
\(\mathrm{F}_4 \supset \mathrm{O}(9) \)
\(\mathrm{E}_6 \supset \mathrm{O}(10) \)
\(\mathrm{E}_7 \supset \mathrm{O}(12) \)
\( \mathrm{E}_8 \supset \mathrm{O}(16) \)

Conformal group
Main article: Conformal group

Being isometries, real orthogonal transforms preserve angles, and are thus conformal maps, though not all conformal linear transforms are orthogonal. In classical terms this is the difference between congruence and similarity, as exemplified by SSS (Side-Side-Side) congruence of triangles and AAA (Angle-Angle-Angle) similarity of triangles. The group of conformal linear maps of Rn is denoted CO(n) for the conformal orthogonal group, and consists of the product of the orthogonal group with the group of dilations. If n is odd, these two subgroups do not intersect, and they are a direct product: CO(2k + 1) = O(2k + 1) × R∗, where R∗ = R∖{0} is the real multiplicative group, while if n is even, these subgroups intersect in ±1, so this is not a direct product, but it is a direct product with the subgroup of dilation by a positive scalar: CO(2k) = O(2k) × R+.

Similarly one can define CSO(n); note that this is always: CSO(n) = CO(n) ∩ GL+(n) = SO(n) × R+.
Discrete subgroups

As the orthogonal group is compact, discrete subgroups are equivalent to finite subgroups.[note 2] These subgroups are known as point group and can be realized as the symmetry groups of polytopes. A very important class of examples are the finite Coxeter groups, which include the symmetry groups of regular polytopes.

Dimension 3 is particularly studied – see point groups in three dimensions, polyhedral groups, and list of spherical symmetry groups. In 2 dimensions, the finite groups are either cyclic or dihedral – see point groups in two dimensions.

Other finite subgroups include:

Permutation matrices (the Coxeter group An)
Signed permutation matrices (the Coxeter group Bn); also equals the intersection of the orthogonal group with the integer matrices.[note 3]

Covering and quotient groups

The orthogonal group is neither simply connected nor centerless, and thus has both a covering group and a quotient group, respectively:

Two covering Pin groups, Pin+(n) → O(n) and Pin−(n) → O(n),
The quotient projective orthogonal group, O(n) → PO(n).

These are all 2-to-1 covers.

For the special orthogonal group, the corresponding groups are:

Spin group, Spin(n) → SO(n),
Projective special orthogonal group, SO(n) → PSO(n).

Spin is a 2-to-1 cover, while in even dimension, PSO(2k) is a 2-to-1 cover, and in odd dimension PSO(2k + 1) is a 1-to-1 cover, i.e., isomorphic to SO(2k + 1). These groups, Spin(n), SO(n), and PSO(n) are Lie group forms of the compact special orthogonal Lie algebra, \mathfrak{so}(n, {\mathbb R}) – Spin is the simply connected form, while PSO is the centerless form, and SO is in general neither.[note 4]

In dimension 3 and above these are the covers and quotients, while dimension 2 and below are somewhat degenerate; see specific articles for details.
Principal homogeneous space: Stiefel manifold
Main article: Stiefel manifold

The principal homogeneous space for the orthogonal group O(n) is the Stiefel manifold Vn(Rn) of orthonormal bases (orthonormal n-frames).

In other words, the space of orthonormal bases is like the orthogonal group, but without a choice of base point: given an orthogonal space, there is no natural choice of orthonormal basis, but once one is given one, there is a one-to-one correspondence between bases and the orthogonal group. Concretely, a linear map is determined by where it sends a basis: just as an invertible map can take any basis to any other basis, an orthogonal map can take any orthogonal basis to any other orthogonal basis.

The other Stiefel manifolds Vk(Rn) for k < n of incomplete orthonormal bases (orthonormal k-frames) are still homogeneous spaces for the orthogonal group, but not principal homogeneous spaces: any k-frame can be taken to any other k-frame by an orthogonal map, but this map is not uniquely determined.
See also
Specific transforms

Coordinate rotations and reflections
Reflection through the origin

Specific groups

rotation group, SO(3, R)
SO(8)

Related groups

indefinite orthogonal group
unitary group
symplectic group

Lists of groups

list of finite simple groups
list of simple Lie groups

Notes

The analogy is stronger: Weyl groups, a class of (representations of) Coxeter groups, can be considered as simple algebraic groups over the field with one element, and there are a number of analogies between algebraic groups and vector spaces on the one hand, and Weyl groups and sets on the other.
Infinite subsets of a compact space have an accumulation point and are not discrete.
O(n) ∩ GL(n, Z) equals the signed permutation matrices because an integer vector of norm 1 must have a single non-zero entry, which must be ±1 (if it has two non-zero entries or a larger entry, the norm will be larger than 1), and in an orthogonal matrix these entries must be in different coordinates, which is exactly the signed permutation matrices.

In odd dimension, SO(2k + 1) ≅ PSO(2k + 1) is centerless (but not simply connected), while in even dimension SO(2k) is neither centerless nor simply connected.

References

For base fields of characteristic not 2, it is equivalent to use bilinear forms or quadratic forms. But in characteristic 2 these notions differ.
John Baez "This Week's Finds in Mathematical Physics" week 105
Wilson, Robert A. (2009). The finite simple groups. Graduate Texts in Mathematics 251. London: Springer. pp. 69–75. ISBN 978-1-84800-987-5. Zbl 1203.20012.
Knus, Max-Albert (1991), Quadratic and Hermitian forms over rings, Grundlehren der Mathematischen Wissenschaften 294, Berlin etc.: Springer-Verlag, p. 224, ISBN 3-540-52117-8, Zbl 0756.11008
(Taylor 1992, page 160)
(Grove 2002, Theorem 6.6 and 14.16)

Cassels 1978, p. 178

Cassels, J.W.S. (1978), Rational Quadratic Forms, London Mathematical Society Monographs 13, Academic Press, ISBN 0-12-163260-1, Zbl 0395.10029
Grove, Larry C. (2002), Classical groups and geometric algebra, Graduate Studies in Mathematics 39, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-2019-3, MR 1859189
Taylor, Donald E. (1992), The Geometry of the Classical Groups, Sigma Series in Pure Mathematics 9, Berlin: Heldermann Verlag, ISBN 3-88538-009-9, MR 1189139, Zbl 0767.20001

External links

Hazewinkel, Michiel, ed. (2001), "Orthogonal group", Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
John Baez "This Week's Finds in Mathematical Physics" week 105
John Baez on Octonions
(Italian) n-dimensional Special Orthogonal Group parametrization

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