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In mathematics, the orthogonal group in dimension , denoted , is the
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
of distance-preserving transformations of a
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
of dimension 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 In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
. Equivalently, it is the group of
orthogonal matrices In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose of and is the identity ma ...
, where the group operation is given by
matrix multiplication In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the ...
(an orthogonal matrix is a
real matrix In mathematics, a matrix (plural matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object. For example, \beg ...
whose inverse equals its
transpose In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations). The t ...
). The orthogonal group is an
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. ...
and a
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the ad ...
. It is
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
. The orthogonal group in dimension has two connected components. The one that contains the
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures ...
is a normal subgroup, called the special orthogonal group, and denoted . It consists of all orthogonal matrices of
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if an ...
. This group is also called the rotation group, generalizing the fact that in dimensions 2 and 3, its elements are the usual
rotations Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
around a point (in dimension 2) or a line (in dimension 3). In low dimension, these groups have been widely studied, see , and . The other component consists of all orthogonal matrices of determinant . 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 , an matrix with entries in such that its inverse equals its transpose is called an ''orthogonal matrix over'' . The orthogonal matrices form a subgroup, denoted , of the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
; that is \operatorname(n, F) = \left\ . More generally, given a non-degenerate
symmetric bilinear form In mathematics, a symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a bilinea ...
or
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...
on a
vector space In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called ''vector (mathematics and physics), vectors'', may be Vector addition, added together and Scalar multiplication, mu ...
over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
, the ''orthogonal group of the form'' is the group of invertible
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
s that preserve the form. The preceding orthogonal groups are the special case where, on some basis, the bilinear form is the
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algeb ...
, or, equivalently, the quadratic form is the sum of the square of the coordinates. All orthogonal groups are
algebraic groups In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Ma ...
, since the condition of preserving a form can be expressed as an equality of matrices.


Name

The name of "orthogonal group" originates from the following characterization of its elements. Given a
Euclidean vector space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
of dimension , the elements of the orthogonal group are,
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' a ...
a
uniform scaling In affine geometry, uniform scaling (or isotropic scaling) is a linear transformation that enlarges (increases) or shrinks (diminishes) objects by a ''scale factor'' that is the same in all directions. The result of uniform scaling is similar ...
(
homothecy In mathematics, a homothety (or homothecy, or homogeneous dilation) is a transformation of an affine space determined by a point ''S'' called its ''center'' and a nonzero number ''k'' called its ''ratio'', which sends point X to a point X' by th ...
), the
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
s from to that map orthogonal vectors to orthogonal vectors.


In Euclidean geometry

The orthogonal group is the subgroup of the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
, consisting of all endomorphisms that preserve the
Euclidean norm Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean s ...
, that is endomorphisms such that \, g(x)\, = \, x\, . Let be the group of the Euclidean isometries of a
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
of dimension . This group does not depend on the choice of a particular space, since all Euclidean spaces of the same dimension are
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
. The
stabilizer subgroup In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
of a point is the subgroup of the elements such that . This stabilizer is (or, more exactly, is isomorphic to) , 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 In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) w ...
from to , which is defined by :p(g)(y-x) = g(y)-g(x), where, as usual, the subtraction of two points denotes the
translation Translation is the communication of the meaning of a source-language text by means of an equivalent target-language text. The English language draws a terminological distinction (which does not exist in every language) between ''transl ...
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 (for details, see ). The
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learni ...
of is the vector space of the translations. So, the translation form a
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
of , the stabilizers of two points are conjugate under the action of the translations, and all stabilizers are isomorphic to . Moreover, the Euclidean group is a
semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in ...
of and the group of translations. It follows that the study of the Euclidean group is essentially reduced to the study of .


Special orthogonal group

By choosing an
orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For exam ...
of a Euclidean vector space, the orthogonal group can be identified with the group (under matrix multiplication) of
orthogonal matrices In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. One way to express this is Q^\mathrm Q = Q Q^\mathrm = I, where is the transpose of and is the identity ma ...
, which are the matrices such that : Q Q^\mathsf = I. It follows from this equation that the square of the
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if an ...
of equals , and thus the determinant of is either or . The orthogonal matrices with determinant form a subgroup called the ''special orthogonal group'', denoted , consisting of all direct isometries of , which are those that preserve the
orientation Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building des ...
of the space. is a normal subgroup of , as being the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learni ...
of the determinant, which is a group homomorphism whose image is the multiplicative group This implies that the orthogonal group is an internal
semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in ...
of and any subgroup formed with the identity and a reflection. The group with two elements (where is the identity matrix) is a
normal subgroup In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
and even a
characteristic subgroup In mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is mapped to itself by every automorphism of the parent group. Because every conjugation map is an inner automorphi ...
of , and, if is even, also of . If is odd, is the internal
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
of and . The group is abelian (this is not the case of for every ). Its finite subgroups are the
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bina ...
of -fold rotations, for every positive integer . All these groups are normal subgroup of and .


Canonical form

For any element of there is an orthogonal basis, where its matrix has the form :\begin \begin R_1 & & \\ & \ddots & \\ & & R_k \end & 0 \\ 0 & \begin \pm 1 & & \\ & \ddots & \\ & & \pm 1 \end\\ \end, where the matrices are 2-by-2 rotation matrices, that is matrices of the form :\begina&b\\-b&a\end, with a^2+b^2=1. This results from the spectral theorem by regrouping
eigenvalues In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
that are
complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...
, and taking into account that the absolute values of the eigenvalues of an orthogonal matrix are all equal to 1. The element belongs to if and only if there are an even number of on the diagonal. The special case of is known as
Euler's rotation theorem In geometry, Euler's rotation theorem states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed ...
, which asserts that every (non-identity) element of is a
rotation Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
about a unique axis-angle pair.


Reflections

Reflections are the elements of whose canonical form is :\begin-1&0\\0&I\end, where is the 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 A mirror image (in a plane mirror) is a reflected duplication of an object that appears almost identical, but is reversed in the direction perpendicular to the mirror surface. As an optical effect it results from reflection off from substances ...
with respect to a
hyperplane In geometry, a hyperplane is a subspace whose dimension is one less than that of its ''ambient space''. For example, if a space is 3-dimensional then its hyperplanes are the 2-dimensional planes, while if the space is 2-dimensional, its hyperp ...
. In dimension two, every rotation is the product of two reflections. More precisely, a rotation of angle is the product of two reflections whose axes have an angle of . Every element of is the product of at most reflections. This results immediately from the above canonical form and the case of dimension two. The
Cartan–Dieudonné theorem In mathematics, the Cartan–Dieudonné theorem, named after Élie Cartan and Jean Dieudonné, establishes that every orthogonal transformation in an ''n''-dimensional symmetric bilinear space can be described as the composition of at most ''n'' ...
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 In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...
(the map ) is an example of an element of that is not the product of fewer than reflections.


Symmetry group of spheres

The orthogonal group is the
symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambie ...
of the -sphere (for , this is just the
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...
) and all objects with spherical symmetry, if the origin is chosen at the center. The
symmetry group In group theory, the symmetry group of a geometric object is the group of all transformations under which the object is invariant, endowed with the group operation of composition. Such a transformation is an invertible mapping of the ambie ...
of a
circle A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre. Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is con ...
is . The orientation-preserving subgroup is isomorphic (as a ''real'' Lie group) to the
circle group In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers. \mathbb T = \. ...
, also known as , the multiplicative group of the
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s of absolute value equal to one. This isomorphism sends the complex number of
absolute value In mathematics, the absolute value or modulus of a real number x, is the non-negative value without regard to its sign. Namely, , x, =x if is a positive number, and , x, =-x if x is negative (in which case negating x makes -x positive), an ...
  to the special orthogonal matrix :\begin \cos(\varphi) & -\sin(\varphi) \\ \sin(\varphi) & \cos(\varphi) \end. In higher dimension, has a more complicated structure (in particular, it is no longer commutative). The
topological In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
structures of the -sphere and are strongly correlated, and this correlation is widely used for studying both
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s.


Group structure

The groups and are real
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the ad ...
s of
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...
. The group has two connected components, with being the
identity component In mathematics, specifically group theory, the identity component of a group ''G'' refers to several closely related notions of the largest connected subgroup of ''G'' containing the identity element. In point set topology, the identity compon ...
, that is, the connected component containing the
identity matrix In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or ...
.


As algebraic groups

The orthogonal group can be identified with the group of the matrices such that A^\mathsfA = I. Since both members of this equation are
symmetric matrices In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with re ...
, this provides \textstyle \frac 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 is an
algebraic set Algebraic may refer to any subject related to algebra in mathematics and related branches like algebraic number theory and algebraic topology. The word algebra itself has several meanings. Algebraic may also refer to: * Algebraic data type, a data ...
. Moreover, it can be proved that its dimension is :\frac = n^2 - \frac, which implies that is a
complete intersection In mathematics, an algebraic variety ''V'' in projective space is a complete intersection if the ideal of ''V'' is generated by exactly ''codim V'' elements. That is, if ''V'' has dimension ''m'' and lies in projective space ''P'n'', there shou ...
. This implies that all its
irreducible component In algebraic geometry, an irreducible algebraic set or irreducible variety is an algebraic set that cannot be written as the union of two proper algebraic subsets. An irreducible component is an algebraic subset that is irreducible and maximal (fo ...
s have the same dimension, and that it has no embedded component. In fact, has two irreducible components, that are distinguished by the sign of the determinant (that is or ). Both are nonsingular algebraic varieties of the same dimension . The component with is .


Maximal tori and Weyl groups

A
maximal torus In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups. A torus in a compact Lie group ''G'' is a compact, connected, abelian Lie subgroup of ''G'' (and therefore ...
in a compact
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the ad ...
''G'' is a maximal subgroup among those that are isomorphic to for some , where is the standard one-dimensional torus. In and , for every maximal torus, there is a basis on which the torus consists of the block-diagonal matrices of the form :\begin R_1 & & 0 \\ & \ddots & \\ 0 & & R_n \end, where each belongs to . In and , the maximal tori have the same form, bordered by a row and a column of zeros, and 1 on the diagonal. The
Weyl group In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections th ...
of is the
semidirect product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in ...
\^n \rtimes S_n of a normal
elementary abelian In mathematics, specifically in group theory, an elementary abelian group (or elementary abelian ''p''-group) is an abelian group in which every nontrivial element has order ''p''. The number ''p'' must be prime, and the elementary abelian group ...
2-subgroup and a
symmetric group In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
, where the nontrivial element of each factor of acts on the corresponding circle factor of by
inversion Inversion or inversions may refer to: Arts * , a French gay magazine (1924/1925) * ''Inversion'' (artwork), a 2005 temporary sculpture in Houston, Texas * Inversion (music), a term with various meanings in music theory and musical set theory * ...
, and the symmetric group acts on both and by permuting factors. The elements of the Weyl group are represented by matrices in . The factor is represented by block permutation matrices with 2-by-2 blocks, and a final 1 on the diagonal. The component is represented by block-diagonal matrices with 2-by-2 blocks either :\begin 1 & 0 \\ 0 & 1 \end \quad \text \quad \begin 0 & 1 \\ 1 & 0 \end, with the last component chosen to make the determinant 1. The Weyl group of is the subgroup H_ \rtimes S_n < \^n \rtimes S_n of that of , where is the
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learni ...
of the product homomorphism given by \left(\epsilon_1, \ldots, \epsilon_n\right) \mapsto \epsilon_1 \cdots \epsilon_n; that is, is the subgroup with an even number of minus signs. The Weyl group of is represented in by the preimages under the standard injection of the representatives for the Weyl group of . Those matrices with an odd number of \begin 0 & 1 \\ 1 & 0 \end blocks have no remaining final coordinate to make their determinants positive, and hence cannot be represented in .


Topology


Low-dimensional topology

The low-dimensional (real) orthogonal groups are familiar
space Space is the boundless three-dimensional extent in which objects and events have relative position and direction. In classical physics, physical space is often conceived in three linear dimensions, although modern physicists usually con ...
s: * , a
two 2 (two) is a number, numeral and digit. It is the natural number following 1 and preceding 3. It is the smallest and only even prime number. Because it forms the basis of a duality, it has religious and spiritual significance in many cultu ...
-point
discrete space In topology, a discrete space is a particularly simple example of a topological space or similar structure, one in which the points form a , meaning they are '' isolated'' from each other in a certain sense. The discrete topology is the finest to ...
* * is * is * is doubly covered by .


Fundamental group

In terms of
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify u ...
, for the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
of is cyclic of order 2, and the
spin group In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1. As a Li ...
is its
universal cover A covering of a topological space X is a continuous map \pi : E \rightarrow X with special properties. Definition Let X be a topological space. A covering of X is a continuous map : \pi : E \rightarrow X such that there exists a discrete sp ...
. For the fundamental group is infinite cyclic and the universal cover corresponds to the
real line In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a poin ...
(the group is the unique connected 2-fold cover).


Homotopy groups

Generally, the
homotopy group In mathematics, homotopy groups are used in algebraic topology to classify topological spaces. The first and simplest homotopy group is the fundamental group, denoted \pi_1(X), which records information about loops in a space. Intuitively, homot ...
s of the real orthogonal group are related to
homotopy groups of spheres In the mathematical field of algebraic topology, the homotopy groups of spheres describe how spheres of various dimensions can wrap around each other. They are examples of topological invariants, which reflect, in algebraic terms, the structure ...
, 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 In mathematics, a direct limit is a way to construct a (typically large) object from many (typically smaller) objects that are put together in a specific way. These objects may be groups, rings, vector spaces or in general objects from any categ ...
of the sequence of inclusions: :\operatorname(0) \subset \operatorname(1)\subset \operatorname(2) \subset \cdots \subset O = \bigcup_^\infty \operatorname(k) Since the inclusions are all closed, hence
cofibration In mathematics, in particular homotopy theory, a continuous mapping :i: A \to X, where A and X are topological spaces, is a cofibration if it lets homotopy classes of maps ,S/math> be extended to homotopy classes of maps ,S/math> whenever a map ...
s, this can also be interpreted as a union. On the other hand, is a homogeneous space for , and one has the following
fiber bundle In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and ...
: : \operatorname(n) \to \operatorname(n + 1) \to S^n, which can be understood as "The orthogonal group acts transitively on the unit sphere , and the stabilizer of a point (thought of as a
unit vector In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vec ...
) is the orthogonal group of the perpendicular complement, which is an orthogonal group one dimension lower." Thus the natural inclusion is -connected, so the homotopy groups stabilize, and for : thus the homotopy groups of the stable space equal the lower homotopy groups of the unstable spaces. From
Bott periodicity In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by , which proved to be of foundational significance for much further research, in particular in K-theory of stable comple ...
we obtain , therefore the homotopy groups of are 8-fold periodic, meaning , and one need only to list the lower 8 homotopy groups: : \begin \pi_0 (O) &= \mathbf/2\mathbf\\ \pi_1 (O) &= \mathbf/2\mathbf\\ \pi_2 (O) &= 0\\ \pi_3 (O) &= \mathbf\\ \pi_4 (O) &= 0\\ \pi_5 (O) &= 0\\ \pi_6 (O) &= 0\\ \pi_7 (O) &= \mathbf \end


Relation to KO-theory

Via the
clutching construction In topology, a branch of mathematics, the clutching construction is a way of constructing fiber bundles, particularly vector bundles on spheres. Definition Consider the sphere S^n as the union of the upper and lower hemispheres D^n_+ and D^n_- alon ...
, homotopy groups of the stable space are identified with stable vector bundles on spheres (
up to isomorphism Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' ...
), with a dimension shift of 1: . Setting (to make fit into the periodicity), one obtains: : \begin \pi_0 (KO) &= \mathbf\\ \pi_1 (KO) &= \mathbf/2\mathbf\\ \pi_2 (KO) &= \mathbf/2\mathbf\\ \pi_3 (KO) &= 0\\ \pi_4 (KO) &= \mathbf\\ \pi_5 (KO) &= 0\\ \pi_6 (KO) &= 0\\ \pi_7 (KO) &= 0 \end


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. *, from
orientation Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building des ...
-preserving/reversing (this class survives to and hence stably) *, which is
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...
comes from . *, which surjects onto ; this latter thus vanishes.


=Lie groups

= From general facts about
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the ad ...
s, always vanishes, and is free ( free abelian).


=Vector bundles

= From the vector bundle point of view, is vector bundles over , 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 is
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...
.


=Loop spaces

= Using concrete descriptions of the loop spaces in
Bott periodicity In mathematics, the Bott periodicity theorem describes a periodicity in the homotopy groups of classical groups, discovered by , which proved to be of foundational significance for much further research, in particular in K-theory of stable comple ...
, one can interpret the higher homotopies of in terms of simpler-to-analyze homotopies of lower order. Using π0, and have two components, and have
countably many In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
components, and the rest are connected.


Interpretation of homotopy groups

In a nutshell: * is about
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...
* is about
orientation Orientation may refer to: Positioning in physical space * Map orientation, the relationship between directions on a map and compass directions * Orientation (housing), the position of a building with respect to the sun, a concept in building des ...
* is about
spin Spin or spinning most often refers to: * Spinning (textiles), the creation of yarn or thread by twisting fibers together, traditionally by hand spinning * Spin, the rotation of an object around a central axis * Spin (propaganda), an intentionally b ...
* is about
topological quantum field theory In gauge theory and mathematical physics, a topological quantum field theory (or topological field theory or TQFT) is a quantum field theory which computes topological invariants. Although TQFTs were invented by physicists, they are also of mathem ...
. Let be any of the four
division algebra In the field of mathematics called abstract algebra, a division algebra is, roughly speaking, an algebra over a field in which division, except by zero, is always possible. Definitions Formally, we start with a non-zero algebra ''D'' over a fiel ...
s , , , , and let be the
tautological line bundle In mathematics, the tautological bundle is a vector bundle occurring over a Grassmannian in a natural tautological way: for a Grassmannian of k- dimensional subspaces of V, given a point in the Grassmannian corresponding to a k-dimensional vector s ...
over the
projective line In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a ''point at infinity''. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; ...
, and its class in K-theory. Noting that , , , , these yield vector bundles over the corresponding spheres, and * is generated by * is generated by * is generated by * is generated by From the point of view of
symplectic geometry Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the H ...
, can be interpreted as the
Maslov index In mathematics, the Lagrangian Grassmannian is the smooth manifold of Lagrangian subspaces of a real symplectic vector space ''V''. Its dimension is ''n''(''n'' + 1) (where the dimension of ''V'' is ''2n''). It may be identified with the homogene ...
, thinking of it as the fundamental group of the stable
Lagrangian Grassmannian In mathematics, the Lagrangian Grassmannian is the smooth manifold of Lagrangian subspaces of a real symplectic vector space ''V''. Its dimension is ''n''(''n'' + 1) (where the dimension of ''V'' is ''2n''). It may be identified with the homogeneous ...
as , so .


Whitehead tower

The orthogonal group anchors a Whitehead tower: :\ldots \rightarrow \operatorname(n) \rightarrow \operatorname(n) \rightarrow \operatorname(n) \rightarrow \operatorname(n) \rightarrow \operatorname(n) which is obtained by successively removing (killing) homotopy groups of increasing order. This is done by constructing
short exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the contex ...
s starting with an Eilenberg–MacLane space for the homotopy group to be removed. The first few entries in the tower are the
spin group In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1. As a Li ...
and the string group, and are preceded by the fivebrane group. The homotopy groups that are killed are in turn 0(''O'') to obtain ''SO'' from ''O'', 1(''O'') to obtain ''Spin'' from ''SO'', 3(''O'') to obtain ''String'' from ''Spin'', and then 7(''O'') and so on to obtain the higher order
brane In string theory and related theories such as supergravity theories, a brane is a physical object that generalizes the notion of a point particle to higher dimensions. Branes are dynamical objects which can propagate through spacetime accord ...
s.


Of indefinite quadratic form over the reals

Over the real numbers,
nondegenerate quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to ...
s are classified by
Sylvester's law of inertia Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remain invariant under a change of basis. Namely, if ''A'' is the symmetric matrix that defines the quadr ...
, which asserts that, on a vector space of dimension , such a form can be written as the difference of a sum of squares and a sum of squares, with . In other words, there is a basis on which the matrix of the quadratic form is a
diagonal matrix In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal m ...
, with entries equal to , and entries equal to . The pair 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 . Moreover, as a quadratic form and its opposite have the same orthogonal group, one has . The standard orthogonal group is . So, in the remainder of this section, it is supposed that neither nor is zero. The subgroup of the matrices of determinant 1 in is denoted . The group 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 . The group is the
Lorentz group In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physic ...
that is fundamental in
relativity theory The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in ...
. Here the corresponds to space coordinates, and corresponds to the time coordinate.


Of complex quadratic forms

Over the field of
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s, every non-degenerate
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...
in variables is equivalent to x_1^2+\cdots+x_n^2. Thus, up to isomorphism, there is only one non-degenerate complex
quadratic space In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to ...
of dimension , and one associated orthogonal group, usually denoted . It is the group of ''complex orthogonal matrices'', complex matrices whose product with their transpose is the identity matrix. As in the real case, has two connected components. The component of the identity consists of all matrices of determinant in ; it is denoted . The groups and are complex Lie groups of dimension over (the dimension over is twice that). For , these groups are noncompact. As in the real case, is not simply connected: For , the
fundamental group In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence classes under homotopy of the loops contained in the space. It records information about the basic shape, or holes, of ...
of is cyclic of order 2, whereas the fundamental group of is .


Over finite fields


Characteristic different from two

Over a field of characteristic different from two, two
quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...
s 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 :''"Witt's theorem" or "the Witt theorem" may also refer to the Bourbaki–Witt fixed point theorem of order theory.'' In mathematics, Witt's theorem, named after Ernst Witt, is a basic result in the algebraic theory of quadratic forms: any isom ...
asserts that (in characteristic different from two) every vector space equipped with a non-degenerate quadratic form can be decomposed as a direct sum of pairwise orthogonal subspaces : V = L_1 \oplus L_2 \oplus \cdots \oplus L_m \oplus W, where each is a
hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...
(that is there is a basis such that the matrix of the restriction of to has the form \textstyle\begin0&1\\1&0\end), and the restriction of to is
anisotropic Anisotropy () is the property of a material which allows it to change or assume different properties in different directions, as opposed to isotropy. It can be defined as a difference, when measured along different axes, in a material's physi ...
(that is, for every nonzero in ). The Chevalley–Warning theorem asserts that, over a
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, sub ...
, the dimension of is at most two. If the dimension of is odd, the dimension of is thus equal to one, and its matrix is congruent either to \textstyle\begin1\end or to \textstyle\begin\varphi\end, where is a non-square scalar. It results that there is only one orthogonal group that is denoted , where is the number of elements of the finite field (a power of an odd prime). If the dimension of is two and is not a square in the ground field (that is, if its number of elements is congruent to 3 modulo 4), the matrix of the restriction of to is congruent to either or , where is the 2×2 identity matrix. If the dimension of is two and is a square in the ground field (that is, if is congruent to 1, modulo 4) the matrix of the restriction of to is congruent to \textstyle\begin1&0\\0&\phi\end, is any non-square scalar. This implies that if the dimension of is even, there are only two orthogonal groups, depending whether the dimension of zero or two. They are denoted respectively and . The orthogonal group is a
dihedral group In mathematics, a dihedral group is the group of symmetries of a regular polygon, which includes rotations and reflections. Dihedral groups are among the simplest examples of finite groups, and they play an important role in group theory, ...
of order , where . When the characteristic is not two, the order of the orthogonal groups are : \left, \operatorname(2n + 1, q)\ = 2q^\prod_^\left(q^ - 1\right), : \left, \operatorname^+(2n, q)\ = 2q^\left(q^n-1\right)\prod_^\left(q^ - 1\right), : \left, \operatorname^-(2n, q)\ = 2q^\left(q^n+ 1\right)\prod_^\left(q^ - 1\right). In characteristic two, the formulas are the same, except that the factor of \left, \operatorname(2n + 1, q)\ must be removed.


The Dickson invariant

For orthogonal groups, the Dickson invariant is a homomorphism from the orthogonal group to the quotient group (integers modulo 2), taking the value in case the element is the product of an even number of reflections, and the value of 1 otherwise. Algebraically, the Dickson invariant can be defined as , where is the identity . 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 Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learni ...
of the Dickson invariant and usually has index 2 in . When the characteristic of is not 2, the Dickson Invariant is whenever the determinant is . Thus when the characteristic is not 2, is commonly defined to be the elements of with determinant . Each element in has determinant . Thus in characteristic 2, the determinant is always . The Dickson invariant can also be defined for
Clifford group In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercom ...
s and
pin group The PIN Group was a German courier and postal services company. It belonged to PIN Group S.A., a Luxembourg-based corporate affiliation made up of several German postal companies. History and shareholding The PIN Group originally traded under ...
s 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. A reflection in characteristic two has a slightly different definition. In characteristic two, the reflection orthogonal to a vector takes a vector to where is the bilinear form and is the quadratic form associated to the orthogonal geometry. Compare this to the
Householder reflection In linear algebra, a Householder transformation (also known as a Householder reflection or elementary reflector) is a linear transformation that describes a reflection about a plane or hyperplane containing the origin. The Householder transformat ...
of odd characteristic or characteristic zero, which takes to . *The
center Center or centre may refer to: Mathematics *Center (geometry), the middle of an object * Center (algebra), used in various contexts ** Center (group theory) ** Center (ring theory) * Graph center, the set of all vertices of minimum eccentricity ...
of the orthogonal group usually has order 1 in characteristic 2, rather than 2, since . *In odd dimensions in characteristic 2, orthogonal groups over
perfect field In algebra, a field ''k'' is perfect if any one of the following equivalent conditions holds: * Every irreducible polynomial over ''k'' has distinct roots. * Every irreducible polynomial over ''k'' is separable. * Every finite extension of ''k'' is ...
s are the same as
symplectic group In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic gr ...
s in dimension . 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 , 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 to the
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For examp ...
(the
multiplicative group In mathematics and group theory, the term multiplicative group refers to one of the following concepts: *the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referre ...
of the field
up to Two mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' a ...
multiplication by
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-lengt ...
elements), that takes reflection in a vector of norm to the image of in . 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 In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups. A Galois group ''G'' associated to a field extension ''L''/''K'' acts in a natu ...
of
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. ...
s, 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 form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...
s over a field can be identified as a Galois , or twisted forms (
torsor In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non- ...
s) 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
determinant In mathematics, the determinant is a scalar value that is a function of the entries of a square matrix. It characterizes some properties of the matrix and the linear map represented by the matrix. In particular, the determinant is nonzero if an ...
. The 'spin' name of the spinor norm can be explained by a connection to the
spin group In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1. As a Li ...
(more accurately a
pin group The PIN Group was a German courier and postal services company. It belonged to PIN Group S.A., a Luxembourg-based corporate affiliation made up of several German postal companies. History and shareholding The PIN Group originally traded under ...
). This may now be explained quickly by Galois cohomology (which however postdates the introduction of the term by more direct use of
Clifford algebra In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hypercom ...
s). The spin covering of the orthogonal group provides a
short exact sequence An exact sequence is a sequence of morphisms between objects (for example, groups, rings, modules, and, more generally, objects of an abelian category) such that the image of one morphism equals the kernel of the next. Definition In the contex ...
of
algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. ...
s. : 1 \rightarrow \mu_2 \rightarrow \mathrm_V \rightarrow \mathrm \rightarrow 1 Here 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 , which is simply the group of -valued points, to is essentially the spinor norm, because is isomorphic to the multiplicative group of the field modulo squares. There is also the connecting homomorphism from of the orthogonal group, to the 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 In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identi ...
corresponding to Lie groups and consists of the skew-symmetric matrices, with the Lie bracket given by the
commutator In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, a ...
. One Lie algebra corresponds to both groups. It is often denoted by \mathfrak(n, F) or \mathfrak(n, F), and called the orthogonal Lie algebra or special orthogonal Lie algebra. Over real numbers, these Lie algebras for different are the compact real forms of two of the four families of
semisimple Lie algebra In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras. (A simple Lie algebra is a non-abelian Lie algebra without any non-zero proper ideals). Throughout the article, unless otherwise stated, a Lie algebra is ...
s: in odd dimension , where , while in even dimension , where . Since the group is not simply connected, the representation theory of the orthogonal Lie algebras includes both representations corresponding to ''ordinary'' representations of the orthogonal groups, and representations corresponding to ''projective'' representations of the orthogonal groups. (The projective representations of are just linear representations of the universal cover, the
spin group In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1. As a Li ...
Spin(''n'').) The latter are the so-called
spin representation In mathematics, the spin representations are particular projective representations of the orthogonal or special orthogonal groups in arbitrary dimension and signature (i.e., including indefinite orthogonal groups). More precisely, they are two equi ...
, which are important in physics. More generally, given a vector space V (over a field with characteristic not equal to 2) with a nondegenerate symmetric bilinear form (\cdot,\cdot), the special orthogonal Lie algebra consists of tracefree endomorphisms \phi which are skew-symmetric for this form ((\phi A, B) + (A, \phi B) = 0). Over a field of characteristic 2 we consider instead the alternating endomorphisms. Concretely we can equate these with the alternating tensors \Lambda^2 V. The correspondence is given by: :v\wedge w \mapsto (v,\cdot)w - (w,\cdot)v This description applies equally for the indefinite special orthogonal Lie algebras \mathfrak(p, q) for symmetric bilinear forms with signature (p,q). Over real numbers, this characterization is used in interpreting the
curl cURL (pronounced like "curl", UK: , US: ) is a computer software project providing a library (libcurl) and command-line tool (curl) for transferring data using various network protocols. The name stands for "Client URL". History cURL was ...
of a vector field (naturally a 2-vector) as an infinitesimal rotation or "curl", hence the name.


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 and 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 space In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, ...
s of independent interest – for example, is the
Lagrangian Grassmannian In mathematics, the Lagrangian Grassmannian is the smooth manifold of Lagrangian subspaces of a real symplectic vector space ''V''. Its dimension is ''n''(''n'' + 1) (where the dimension of ''V'' is ''2n''). It may be identified with the homogeneous ...
.


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(n) \supset \mathrm(n - 1) – preserve an axis :\mathrm(2n) \supset \mathrm(n) \supset \mathrm(n) – are those that preserve a compatible complex structure ''or'' a compatible symplectic structure – see 2-out-of-3 property; also preserves a complex orientation. :\mathrm(2n) \supset \mathrm(n) :\mathrm(7) \supset \mathrm_2


Lie supergroups

The orthogonal group is also an important subgroup of various Lie groups: :\begin \mathrm(n) &\supset \mathrm(n) \\ \mathrm(2n) &\supset \mathrm(n) \\ \mathrm_2 &\supset \mathrm(3) \\ \mathrm_4 &\supset \mathrm(9) \\ \mathrm_6 &\supset \mathrm(10) \\ \mathrm_7 &\supset \mathrm(12) \\ \mathrm_8 &\supset \mathrm(16) \end


Conformal group

Being
isometries In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...
, real orthogonal transforms preserve
angle In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle. Angles formed by two rays lie in the plane that contains the rays. Angles ...
s, and are thus
conformal map In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\i ...
s, 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 Congruence of triangles may refer to: * Congruence (geometry)#Congruence of triangles * Solution of triangles {{disambiguation ...
and AAA (angle-angle-angle) similarity of triangles. The group of conformal linear maps of is denoted for the conformal orthogonal group, and consists of the product of the orthogonal group with the group of dilations. If is odd, these two subgroups do not intersect, and they are a
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
: , where is the real
multiplicative group In mathematics and group theory, the term multiplicative group refers to one of the following concepts: *the group under multiplication of the invertible elements of a field, ring, or other structure for which one of its operations is referre ...
, while if is even, these subgroups intersect in , so this is not a direct product, but it is a direct product with the subgroup of dilation by a positive scalar: . Similarly one can define ; note that this is always: .


Discrete subgroups

As the orthogonal group is compact, discrete subgroups are equivalent to finite subgroups.Infinite subsets of a compact space have an
accumulation point In mathematics, a limit point, accumulation point, or cluster point of a set S in a topological space X is a point x that can be "approximated" by points of S in the sense that every neighbourhood of x with respect to the topology on X also conta ...
and are not discrete.
These subgroups are known as
point group In geometry, a point group is a mathematical group of symmetry operations ( isometries in a Euclidean space) that have a fixed point in common. The coordinate origin of the Euclidean space is conventionally taken to be a fixed point, and ever ...
s and can be realized as the symmetry groups of
polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...
s. A very important class of examples are the
finite Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refl ...
s, which include the symmetry groups of
regular polytope In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...
s. Dimension 3 is particularly studied – see
point groups in three dimensions In geometry, a point group in three dimensions is an isometry group in three dimensions that leaves the origin fixed, or correspondingly, an isometry group of a sphere. It is a subgroup of the orthogonal group O(3), the group of all isometries th ...
, polyhedral groups, and
list of spherical symmetry groups Finite spherical symmetry groups are also called point groups in three dimensions. There are five fundamental symmetry classes which have triangular fundamental domains: dihedral, cyclic, tetrahedral, octahedral, and icosahedral symmetry. This ...
. In 2 dimensions, the finite groups are either cyclic or dihedral – see
point groups in two dimensions In geometry, a two-dimensional point group or rosette group is a group of geometric symmetries ( isometries) that keep at least one point fixed in a plane. Every such group is a subgroup of the orthogonal group O(2), including O(2) itself. Its ...
. Other finite subgroups include: *
Permutation matrices In mathematics, particularly in matrix theory, a permutation matrix is a square binary matrix that has exactly one entry of 1 in each row and each column and 0s elsewhere. Each such matrix, say , represents a permutation of elements and, when ...
(the
Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refl ...
) *
Signed permutation matrices In mathematics, a generalized permutation matrix (or monomial matrix) is a matrix with the same nonzero pattern as a permutation matrix, i.e. there is exactly one nonzero entry in each row and each column. Unlike a permutation matrix, where the no ...
(the
Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refl ...
); also equals the intersection of the orthogonal group with the
integer matrices In mathematics, an integer matrix is a matrix whose entries are all integers. Examples include binary matrices, the zero matrix, the matrix of ones, the identity matrix, and the adjacency matrices used in graph theory, amongst many others. Intege ...
. equals the signed permutation matrices because an integer vector of norm 1 must have a single non-zero entry, which must be (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.


Covering and quotient groups

The orthogonal group is neither
simply connected In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
nor centerless, and thus has both a covering group and a
quotient group A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For examp ...
, respectively: * Two covering
Pin group The PIN Group was a German courier and postal services company. It belonged to PIN Group S.A., a Luxembourg-based corporate affiliation made up of several German postal companies. History and shareholding The PIN Group originally traded under ...
s, and , * The quotient
projective orthogonal group In projective geometry and linear algebra, the projective orthogonal group PO is the induced action of the orthogonal group of a quadratic space ''V'' = (''V'',''Q'')A quadratic space is a vector space ''V'' together with a quadratic form ''Q''; ...
, . These are all 2-to-1 covers. For the special orthogonal group, the corresponding groups are: *
Spin group In mathematics the spin group Spin(''n'') page 15 is the double cover of the special orthogonal group , such that there exists a short exact sequence of Lie groups (when ) :1 \to \mathrm_2 \to \operatorname(n) \to \operatorname(n) \to 1. As a Li ...
, , *
Projective special orthogonal group In projective geometry and linear algebra, the projective orthogonal group PO is the induced action of the orthogonal group of a quadratic space ''V'' = (''V'',''Q'')A quadratic space is a vector space ''V'' together with a quadratic form ''Q''; th ...
, . Spin is a 2-to-1 cover, while in even dimension, is a 2-to-1 cover, and in odd dimension is a 1-to-1 cover; i.e., isomorphic to . These groups, , , and are Lie group forms of the compact special orthogonal Lie algebra, \mathfrak(n, \mathbb) – Spin is the simply connected form, while PSO is the centerless form, and SO is in general neither.In odd dimension, is centerless (but not simply connected), while in even dimension is neither centerless nor simply connected. 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

The
principal homogeneous space In mathematics, a principal homogeneous space, or torsor, for a group ''G'' is a homogeneous space ''X'' for ''G'' in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group ''G'' is a non- ...
for the orthogonal group is the
Stiefel manifold In mathematics, the Stiefel manifold V_k(\R^n) is the set of all orthonormal ''k''-frames in \R^n. That is, it is the set of ordered orthonormal ''k''-tuples of vectors in \R^n. It is named after Swiss mathematician Eduard Stiefel. Likewise one ca ...
of
orthonormal bases In mathematics, particularly linear algebra, an orthonormal basis for an inner product space ''V'' with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example ...
(orthonormal -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 for of ''incomplete'' orthonormal bases (orthonormal -frames) are still homogeneous spaces for the orthogonal group, but not ''principal'' homogeneous spaces: any -frame can be taken to any other -frame by an orthogonal map, but this map is not uniquely determined.


See also


Specific transforms

*
Coordinate rotations and reflections In geometry, two-dimensional rotations and reflections are two kinds of Euclidean plane isometries which are related to one another. A rotation in the plane can be formed by composing a pair of reflections. First reflect a point ''P'' to its ...
*
Reflection through the origin In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...


Specific groups

*rotation group, SO(3, R) * SO(8)


Related groups

*
indefinite orthogonal group In mathematics, the indefinite orthogonal group, is the Lie group of all linear transformations of an ''n''- dimensional real vector space that leave invariant a nondegenerate, symmetric bilinear form of signature , where . It is also called th ...
*
unitary group In mathematics, the unitary group of degree ''n'', denoted U(''n''), is the group of unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group is ...
*
symplectic group In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic gr ...


Lists of groups

*
list of finite simple groups A ''list'' is any set of items in a row. List or lists may also refer to: People * List (surname) Organizations * List College, an undergraduate division of the Jewish Theological Seminary of America * SC Germania List, German rugby union ...
*
list of simple Lie groups In mathematics, a simple Lie group is a connected non-abelian Lie group ''G'' which does not have nontrivial connected normal subgroups. The list of simple Lie groups can be used to read off the list of simple Lie algebras and Riemannian symm ...


Representation theory

* Representations of classical Lie groups *
Brauer algebra In mathematics, a Brauer algebra is an associative algebra introduced by Richard Brauer in the context of the representation theory of the orthogonal group. It plays the same role that the symmetric group does for the representation theory of the ge ...


Notes


Citations


References

* * * *


External links

*
John Baez "This Week's Finds in Mathematical Physics" week 105
* {{in lang, it}
n-dimensional Special Orthogonal Group parametrization
Lie groups Quadratic forms Euclidean symmetries Linear algebraic groups