 Orthogonal group

Group theory Group theory Cyclic group Z_{n}
Symmetric group, S_{n}
Dihedral group, D_{n}
Alternating group A_{n}
Mathieu groups M_{11}, M_{12}, M_{22}, M_{23}, M_{24}
Conway groups Co_{1}, Co_{2}, Co_{3}
Janko groups J_{1}, J_{2}, J_{3}, J_{4}
Fischer groups F_{22}, F_{23}, F_{24}
Baby Monster group B
Monster group MSolenoid (mathematics)
Circle group
General linear group GL(n)
Special linear group SL(n)
Orthogonal group O(n)
Special orthogonal group SO(n)
Unitary group U(n)
Special unitary group SU(n)
Symplectic group Sp(n)
Lorentz group
Poincaré group
Conformal group
Diffeomorphism group
Loop group
Infinitedimensional Lie groups O(∞) SU(∞) Sp(∞)v · General linear group GL(n)
Special linear group SL(n)
Orthogonal group O(n)
Special orthogonal group SO(n)
Unitary group U(n)
Special unitary group SU(n)
Symplectic group Sp(n)Exponential map
Adjoint representation of a Lie group
Adjoint representation of a Lie algebra
Killing form
Lie point symmetryStructure of semisimple Lie groupsDynkin diagrams
Cartan subalgebra
Root system
Real form
Complexification
Split Lie algebra
Compact Lie algebraRepresentation of a Lie group
Representation of a Lie algebrav · mathematics, the orthogonal group of degree n over a field F (written as O(n,F)) is the group 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 where Q^{T} is the transpose of Q. The classical orthogonal group over the real numbers is usually just written O(n).
More generally the orthogonal group of a nonsingular quadratic form over F is the group of linear operators preserving the form – the above group O(n, F) is then the orthogonal group of the sumofnsquares quadratic form.^{[1]} The Cartan–Dieudonné theorem describes the structure of the orthogonal group for nonsingular form. This article only discusses definite forms – the orthogonal group of the positive definite form (equivalent to sum of n squares) and negative definite forms (equivalent to the negative sum of n squares) are identical – O(n,0) = O(0,n) – though the associated Pin groups differ; for other nonsingular forms O(p,q), see indefinite orthogonal group.
Every orthogonal matrix has determinant either 1 or −1. The orthogonal nbyn matrices with determinant 1 form a normal subgroup of O(n,F) known as the special orthogonal group, SO(n,F). (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 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 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.
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.
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 R^{n}; 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 hypersphere 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 – It is the rotation group of the sphere and all objects with spherical symmetry, if the origin is chosen at the center.
{ I, −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 direct product of SO(n,R) and { I, −I }. The cyclic group of kfold rotations C_{k} is for every positive integer k a normal subgroup of O(2,R) and SO(2,R).
Relative to suitable orthogonal bases, the isometries are of the form:
where the matrices R_{1},...,R_{k} are 2by2 rotation matrices in orthogonal planes of rotation. As a special case, known as Euler's rotation theorem, any (nonidentity) 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 ), though so are other maximal combinations of rotations (and a reflection, in odd dimension).
The symmetry group of a circle is O(2,R), also called Dih (S^{1}), where S^{1} denotes the multiplicative group of complex numbers of absolute value 1.
SO(2,R) is isomorphic (as a Lie group) to the circle S^{1} (circle group). This isomorphism sends the complex number exp(φi) = cos(φ) + i sin(φ) to the orthogonal matrix
The group SO(3,R), understood as the set of rotations of 3dimensional space, is of major importance in the sciences and engineering. See rotation group and the general formula for a 3 × 3 rotation matrix in terms of the axis and the angle.
In terms of algebraic topology, for n > 2 the fundamental group of SO(n,R) is cyclic of order 2, and the spinor 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 spinor group Spin(2) is the unique 2fold cover).
Even and odd dimension
The structure of the orthogonal group differs in certain respects between even and odd dimensions – for example, − I (reflection through the origin) is orientationpreserving in even dimension, but orientationreversing in odd dimension. 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 B_{r} = so_{2r + 1}, while in even dimension the Lie algebra is D_{r} = so_{2r}.
Lie algebra
The Lie algebra associated to the Lie groups O(n,R) and SO(n,R) consists of the skewsymmetric real nbyn matrices, with the Lie bracket given by the commutator. This Lie algebra is often denoted by o(n,R) or by so(n,R), and called the orthogonal Lie algebra or special orthogonal Lie algebra. These Lie algebras are the compact real forms of two of the four families of semisimple Lie algebras: in odd dimension B_{r} = so_{2r + 1}, while in even dimension D_{r} = so_{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 (2blades) . The correspondence is given by the map where v ^{*} is the covector dual to the vector v; in coordinates these are exactly the elementary skewsymmetric matrices.
This characterization is used in interpreting the curl of a vector field (naturally a 2vector) as an infinitesimal rotation or "curl", hence the name. Generalizing the inner product with a nondegenerate form yields the indefinite orthogonal Lie algebras 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 socalled spin representation, which are important in physics.
3D isometries that leave the origin fixed
Isometries of R^{3} that leave the origin fixed, forming the group O(3,R), can be categorized as:
 SO(3,R):
 identity
 rotation about an axis through the origin by an angle not equal to 180°
 rotation about an axis through the origin by an angle of 180°
 the same with inversion in the origin (x is mapped to −x), i.e. respectively:
 inversion in the origin
 rotation about an axis by an angle not equal to 180°, combined with reflection in the plane through the origin perpendicular to the axis
 reflection in a plane through the origin.
The 4th and 5th in particular, and in a wider sense the 6th also, are called improper rotations.
See also the similar overview including translations.
Conformal group
Main article: Conformal groupBeing isometries (preserving distances), orthogonal transforms also 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 (SideSideSide) congruence of triangles and AAA (AngleAngleAngle) similarity of triangles. The group of conformal linear maps of R^{n} 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: , while if n 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 CSO(n); note that this is always :.
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 (which 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.
The complex Lie algebra associated to O(n,C) and SO(n,C) consists of the skewsymmetric complex nbyn matrices, with the Lie bracket given by the commutator.
Topology
Low dimensional
The low dimensional (real) orthogonal groups are familiar spaces:
The group SO(4) is double covered by .
There are numerous charts on SO(3), due to the importance of 3dimensional rotations in engineering applications.
Here S^{n} denotes the ndimensional sphere, RP^{n} the ndimensional real projective space, and SU(n) the special unitary group of degree n.
Homotopy groups
The homotopy groups of the 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
(as the inclusions are all closed inclusions, hence cofibrations, this can also be interpreted as a union).
S^{n} is a homogeneous space for O(n + 1), and one has the following fiber bundle:
which can be understood as "The orthogonal group O(n + 1) acts transitively on the unit sphere S^{n}, 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". The map is the natural inclusion.
Thus the inclusion is (n − 1)connected, so the homotopy groups stabilize, and π_{k}(O) = π_{k}(O(n)) for n > k + 1: thus the homotopy groups of the stable space equal the lower homotopy groups of the unstable spaces.
Via Bott periodicity, , thus the homotopy groups of O are 8fold periodic, meaning π_{k + 8}O = π_{k}O, and one needs only to compute the lower 8 homotopy groups to compute them all.
Relation to KOtheory
Via the clutching construction, homotopy groups of the stable space O are identified with stable vector bundles on spheres (up to isomorphism), with a dimension shift of 1: π_{k}O = π_{k + 1}BO.
Setting (to make π_{0} fit into the periodicity), one obtains:
Computation and Interpretation of homotopy groups
Lowdimensional groups
The first few homotopy groups can be calculated by using the concrete descriptions of lowdimensional groups.
 from orientationpreserving/reversing (this class survives to O(2) and hence stably)
yields
 , which is spin
 π_{2}(O) = π_{2}(SO(3)) = 0, which surjects onto π_{2}(SO(4)); this latter thus vanishes.
Lie groups
From general facts about Lie groups, π_{2}G always vanishes, and π_{3}G is free (free abelian).
Vector bundles
From the vector bundle point of view, π_{0}(KO) is vector bundles over S^{0}, which is two points. Thus over each point, the bundle is trivial, and the nontriviality of the bundle is the difference between the dimensions of the vector spaces over the two points, so
 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 π_{0}, O and O/U have two components, and have components, and the rest are connected.
Interpretation of homotopy groups
In a nutshell:^{[2]}
 is dimension
 is orientation
 is spin
 is topological quantum field theory.
Let , and let L_{F} be the tautological line bundle over the projective line , and [L_{F}] its class in Ktheory. Noting that , these yield vector bundles over the corresponding spheres, and
 π_{1}(KO) is generated by
 π_{2}(KO) is generated by
 π_{4}(KO) is generated by
 π_{8}(KO) is generated by
From the point of view of symplectic geometry, can be interpreted as the Maslov index, thinking of it as the fundamental group of the stable Lagrangian Grassmannian π_{1}(U / O), as so π_{1}(U / O) = π_{1 + 7}(KO).
Over finite fields
Orthogonal groups can also be defined over finite fields , where q is a power of a prime p. When defined over such fields, they come in two types in even dimension: O ^{+} (2n,q) and O ^{−} (2n,q); and one type in odd dimension: O(2n + 1,q).
If V is the vector space on which the orthogonal group G acts, it can be written as a direct orthogonal sum as follows:
where L_{i} are hyperbolic lines and W contains no singular vectors. If W = 0, then G is of plus type. If W = < w > then G has odd dimension. If W has dimension 2, G is of minus type.
In the special case where n = 1, is a dihedral group of order .
We have the following formulas for the order of these groups, O(n,q) = { A in GL(n,q) : A·A^{t}=I }, when the characteristic is greater than two
If −1 is a square in
If −1 is a nonsquare in
The Dickson invariant
For orthogonal groups, the Dickson invariant is a homomorphism from the orthogonal group to Z/2Z, and is 0 or 1 depending on whether an element is the product of an even or odd number of reflections. More concretely, the Dickson invariant can be defined as 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 no extra information.
The special orthogonal group is the kernel of the Dickson invariant and usually has index 2 in O(n,F).^{[3]} 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 or 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 behave differently. This section lists some of the differences. Traditionally these groups are known as the hypoabelian groups but this term is no longer used for these groups.
 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.^{[4]} 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
 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}.
For the usual orthogonal group over the reals it is trivial, but it is often nontrivial 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 H^{1}, or twisted forms (torsors) of an orthogonal group. As an algebraic group, an orthogonal group is in general neither connected nor simplyconnected; 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.
Here μ_{2} is the algebraic group of square roots of 1; over a field of characteristic not 2 it is roughly the same as a twoelement group with trivial Galois action. The connecting homomorphism from H^{0}(O_{V}), which is simply the group O_{V}(F) of Fvalued points, to H^{1}(μ_{2}) is essentially the spinor norm, because H^{1}(μ_{2}) is isomorphic to the multiplicative group of the field modulo squares.
There is also the connecting homomorphism from H^{1} of the orthogonal group, to the H^{2} of the kernel of the spin covering. The cohomology is nonabelian, so that this is as far as we can go, at least with the conventional definitions.
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 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:
 – preserves an axis
 – U(n) are those that preserve a compatible complex structure or a compatible symplectic structure – see 2outof3 property; SU(n) also preserves a complex orientation.
Lie supergroups
The orthogonal group O(n) is also an important subgroup of various Lie groups:
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 A_{n})
 Signed permutation matrices (the Coxeter group B_{n}); 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 2to1 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 2to1 cover, while in even dimension, PSO(2k) is a 2to1 cover, and in odd dimension PSO(2k+1) is a 1to1 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, – 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.
Applications to string theory
The group O(10) is of special importance in superstring theory because it is the symmetry group of 10 dimensional spacetime.
Principal homogeneous space: Stiefel manifold
Main article: Stiefel manifoldThe principal homogeneous space for the orthogonal group O(n) is the Stiefel manifold of orthonormal bases (orthonormal nframes).
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 onetoone 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 k < n of incomplete orthonormal bases (orthonormal kframes) are still homogeneous spaces for the orthogonal group, but not principal homogeneous spaces: any kframe can be taken to any other kframe 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
Lists of 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.
 ^ equals the signed permutation matrices because an integer vector of norm 1 must have a single nonzero entry, which must be ±1 (if it has two nonzero 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
 ^ Away from 2, it is equivalent to use bilinear forms or quadratic forms, but at 2 these differ – notably in characteristic 2, but also when generalizing to rings where 2 is not invertible, most significantly the integers, where the notions of even and odd quadratic forms arise.
 ^ John Baez "This Week's Finds in Mathematical Physics" week 105
 ^ (Taylor 1992, page 160)
 ^ (Grove 2002, Theorem 6.6 and 14.16)
 Grove, Larry C. (2002), Classical groups and geometric algebra, Graduate Studies in Mathematics, 39, Providence, R.I.: American Mathematical Society, ISBN 9780821820193, MR1859189
 Taylor, Donald E. (1992), The Geometry of the Classical Groups, 9, Berlin: Heldermann Verlag, ISBN 3885380099, MR1189139
External links
 John Baez "This Week's Finds in Mathematical Physics" week 105
 John Baez on Octonions
 (Italian) ndimensional Special Orthogonal Group parametrization
Categories: Lie groups
 Quadratic forms
 Euclidean symmetries
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Orthogonal group
 Orthogonal group

Group theory Group theory Cyclic group Z_{n}
Symmetric group, S_{n}
Dihedral group, D_{n}
Alternating group A_{n}
Mathieu groups M_{11}, M_{12}, M_{22}, M_{23}, M_{24}
Conway groups Co_{1}, Co_{2}, Co_{3}
Janko groups J_{1}, J_{2}, J_{3}, J_{4}
Fischer groups F_{22}, F_{23}, F_{24}
Baby Monster group B
Monster group MSolenoid (mathematics)
Circle group
General linear group GL(n)
Special linear group SL(n)
Orthogonal group O(n)
Special orthogonal group SO(n)
Unitary group U(n)
Special unitary group SU(n)
Symplectic group Sp(n)
Lorentz group
Poincaré group
Conformal group
Diffeomorphism group
Loop group
Infinitedimensional Lie groups O(∞) SU(∞) Sp(∞)General linear group GL(n)
Special linear group SL(n)
Orthogonal group O(n)
Special orthogonal group SO(n)
Unitary group U(n)
Special unitary group SU(n)
Symplectic group Sp(n)Exponential map
Adjoint representation of a Lie group
Adjoint representation of a Lie algebra
Killing form
Lie point symmetryStructure of semisimple Lie groupsDynkin diagrams
Cartan subalgebra
Root system
Real form
Complexification
Split Lie algebra
Compact Lie algebraRepresentation of a Lie group
Representation of a Lie algebra