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| {{Group theory sidebar |Topological}}
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| In [[mathematics]], the '''orthogonal group''' or '''rotation group'''<ref>{{Mathworld|title = Rotation Group|id=RotationGroup}}</ref> is the [[Group (mathematics)|group]] of [[isometry|distance-preserving transformations]] of [[Euclidean space]] which preserve the [[Origin (mathematics)|origin]], where the group operation is given by [[Function composition|composing]] transformations. Equivalently, it is the group of [[orthogonal matrix|orthogonal matrices]] of a given dimension, where the group operation is given by [[matrix multiplication]]. An orthogonal matrix is a [[real number|real]] matrix whose [[invertible matrix|inverse]] equals its [[transpose]]. The term "orthogonal group" may also refer to a generalization of the above case: the group of invertible [[linear operator]]s which preserve a non-degenerate [[symmetric bilinear form]] or [[quadratic form]]<ref>For base fields of [[Characteristic (algebra)|characteristic]] not 2, it is equivalent to use [[bilinear form]]s or [[quadratic form]]s. But in characteristic 2 these notions differ.</ref> on a [[vector space]] over a [[field (mathematics)|field]].
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| In particular, when the bilinear form is the [[scalar product]] on the vector space {{math|''F<sup> n</sup>''}} of dimension {{mvar|n}} over a [[field (mathematics)|field]] {{mvar|F}}, with quadratic form the sum of squares, then the corresponding orthogonal group, written as {{math|O(''n'', ''F'' )}}, is the set of {{math|''n'' × ''n''}} [[orthogonal matrix|orthogonal matrices]] with entries from {{mvar|F}}, with the group operation of [[matrix multiplication]]. This is a [[subgroup]] of the [[general linear group]] {{math|GL(''n'', ''F'' )}} given by
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| :<math>\mathrm{O}(n,F) = \{ Q \in \mathrm{GL}(n, F) \mid Q^\mathsf{T} Q = Q Q^\mathsf{T} = I \}</math>
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| where {{mvar|Q}}<sup>T</sup> is the [[transpose]] of {{mvar|Q}} and {{math|''I''}} is the [[identity matrix]]. The classical orthogonal group over the real numbers is usually just written {{math|O(''n'')}}. This characterization can be derived as follows: let {{mvar|b}} be a bilinear form, and let <math>Q_i^j</math> be the components of a linear transformation with respect to some basis {{math|{''e<sub>i</sub>''} }}. We may, in fact, choose {{math|{''e<sub>i</sub>''}}} to be such that {{math|''b''(''e<sub>i</sub>'', ''e<sub>j</sub>'')}} is 1 if {{math|1=''i'' = ''j''}} and 0 otherwise, using the [[Gram–Schmidt process]]. Then in order for {{mvar|Q}} to preserve {{mvar|b}}, for any {{math|''e<sub>i</sub>'', ''e<sub>j</sub>''}}, we must have
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| :<math>\begin{align}b(e_i,e_j) &= b(Q(e_i),Q(e_j))\\ &= b(\sum_{k} Q^k_i e_k, \sum_{n}Q^n_j e_n)\\ &= \sum_{n,k} Q^k_i Q^n_j b(e_k,e_n)\\ &= \sum_{k} Q^k_i Q^k_j \end{align} </math>
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| The final term is just the {{math|(''i'', ''j'')}} component of {{mvar|Q}} {{mvar|Q}}<sup>T</sup>, which must therefore be 1 on the diagonal and 0 off it. This implies that {{mvar|Q}} {{mvar|Q}}<sup>T</sup> = {{mvar|I}}. Conversely, if {{mvar|Q}} preserves the form for basis vectors, it is easy to check that by linearity, {{mvar|Q}} preserves it for all vectors.
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| This article mainly discusses [[Definite quadratic form|''definite'' forms]]: the orthogonal group of the positive definite form (equivalent to the sum of {{mvar|n}} squares). Negative definite forms (equivalent to the negative sum of {{mvar|n}} squares) are identical since {{math|1=O(''n'', 0) = O(0, ''n'')}}. However, the associated [[Pin group]]s differ.
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| For {{math|O(''p'', ''q'')}} of other non-singular forms, see [[indefinite orthogonal group]].
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| The [[Commutator subgroup|derived subgroup]] {{math|Ω(''n'', ''F'' )}} of {{math|O(''n'', ''F'')}} is an often studied object because, when {{mvar|F}} is a [[finite field]], {{math|Ω(''n'', ''F'' )}} is often{{clarify|date=February 2013}} a central extension of a finite simple group.
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| Both {{math|O(''n'', ''F'' )}} and {{math|SO(''n'', ''F'' )}} are [[algebraic group]]s, because the condition that a matrix be orthogonal, i.e. have its own [[transpose]] as [[inverse matrix|inverse]], can be expressed as a set of polynomial equations in the entries of the matrix. The [[Cartan–Dieudonné theorem]] describes the structure of the orthogonal group for a non-singular form.
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| ==Name==
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| The [[determinant]] of any orthogonal matrix is either 1 or −1. The orthogonal {{mvar|n}}-by-{{mvar|n}} matrices with determinant 1 form a [[normal subgroup]] of {{math|O(''n'', ''F'' )}} known as the '''special orthogonal group''' {{math|SO(''n'', ''F'' )}}, consisting of all [[proper rotation]]s. (More precisely, {{math|SO(''n'', ''F'' )}} is the [[Kernel (matrix)|kernel]] of the [[Orthogonal group#The Dickson invariant|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 {{math|O(''p'', ''q'')}}. The term '''rotation group''' can be used to describe either the special or general orthogonal group.
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| == Even and odd dimension ==
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| The structure of the orthogonal group differs in certain aspects between even and odd dimensions – for example, over [[ordered field]]s (such as [[real number|{{math|'''R'''}}]]) the [[reflection through the origin|{{math|−''I''}} element]] is [[orientation (vector space)|orientation]]-preserving in even dimensions, but orientation-reversing in odd dimensions. When this distinction wishes to be emphasized, the groups are generally denoted {{math|O(2''k'')}} and {{math|O(2''k'' + 1)}}, reserving {{mvar|n}} for the dimension of the space ({{math|1=''n'' = 2''k''}} or {{math|1=''n'' = 2''k'' + 1}}). The letters {{mvar|p}} or {{mvar|r}} are also used, indicating the [[Rank (Lie algebra)|rank]] of [[#Lie algebra|the corresponding Lie algebra]]; in odd dimension the corresponding Lie algebra is <math>\mathfrak{so}(2r + 1)</math>, while in even dimension the Lie algebra is <math>\mathfrak{so}(2r)</math>.
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| == Over the real number field ==
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| Over the field {{math|'''R'''}} of [[real number]]s, the orthogonal group {{math|O(''n'', '''R''')}} and the special orthogonal group {{math|SO(''n'', '''R''')}} are often simply denoted by {{math|O(''n'')}} and {{math|SO(''n'')}} if no confusion is possible. They form real [[Compact space|compact]] [[Lie group]]s of [[dimension (mathematics)|dimension]] {{math|''n''(''n'' − 1)/2}}. {{math|O(''n'', '''R''')}} has two [[Connected space|connected components]], with {{math|SO(''n'', '''R''')}} being the [[identity component]], i.e., the connected component containing the [[identity matrix]].
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| === Geometric interpretation ===
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| The real orthogonal and real special orthogonal groups have the following geometric interpretations:
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| {{math|O(''n'', '''R''')}} is a subgroup of the [[Euclidean group]] {{math|''E''(''n'')}}, the group of [[isometry|isometries]] of {{math|'''R'''<sup>''n''</sup>}}; it contains those that leave the origin fixed – {{math|1=O(''n'', '''R''') = ''E''(''n'') ∩ GL(''n'', '''R''')}}. It is the symmetry group of the [[sphere]] ({{math|1=''n'' = 3}}) or [[n-sphere|{{math|(''n'' − 1)}}-sphere]] and all objects with spherical symmetry, if the origin is chosen at the center.
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| {{math|SO(''n'', '''R''')}} is a subgroup of {{math|''E''<sup>+</sup>(''n'')}}, which consists of [[Euclidean group#Direct and indirect isometries|''direct'' isometries]], i.e., isometries preserving [[orientation (geometry)|orientation]]; it contains those that leave the origin fixed – {{math|1=SO(''n'', '''R''') = ''E''<sup>+</sup>(''n'') ∩ GL(''n'', '''R''') = ''E''(''n'') ∩ GL<sup>+</sup>(''n'', '''R''')}}. It is the rotation group of the sphere and all objects with spherical symmetry, if the origin is chosen at the center.
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| {{math|{±''I''}}} is a [[normal subgroup]] and even a [[characteristic subgroup]] of {{math|O(''n'', '''R''')}}, and, if {{mvar|n}} is even, also of {{math|SO(''n'', '''R''')}}. If {{mvar|n}} is odd, {{math|O(''n'', '''R''')}} is the internal [[direct product of groups|direct product]] of {{math|SO(''n'', '''R''')}} and {{math|{±''I''}}}. For every positive integer {{mvar|k}} the [[cyclic group]] {{math|''C''<sub>''k''</sub>}} of [[rotational symmetry|{{mvar|k}}-fold rotations]] is a normal subgroup of {{math|O(2, '''R''')}} and {{math|SO(2, '''R''')}}.
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| Relative to suitable orthogonal bases, the isometries are of the form:
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| :<math>\begin{bmatrix}
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| \begin{matrix}R_1 & & \\ & \ddots & \\ & & R_k\end{matrix} & 0 \\
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| 0 & \begin{matrix}\pm 1 & & \\ & \ddots & \\ & & \pm 1\end{matrix} \\
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| \end{bmatrix}</math>
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| where the matrices {{math|''R''<sub>1</sub>, … , ''R<sub>k</sub>''}} are 2-by-2 rotation matrices in orthogonal [[planes of rotation]]. As a special case, known as [[Euler's rotation theorem]], any (non-identity) element of {{math|SO(3, '''R''')}} is [[rotation]] about a uniquely defined axis.
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| The orthogonal group is generated by reflections ([[Coordinate rotations and reflections|two reflections give a rotation]]), as in a [[Coxeter group]],<ref group="note">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.</ref> and elements have [[length function|length]] at most {{mvar|n}} (require at most {{mvar|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 group]]s, by the [[Cartan–Dieudonné theorem]]). A [[Longest element of a Coxeter group|longest element]] (element needing the most reflections) is [[reflection through the origin]] (the map {{math|''v'' ↦ −''v''}}), though so are other maximal combinations of rotations (and a reflection, in odd dimension).
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| The [[symmetry group]] of a [[circle]] is {{math|O(2, '''R''')}}. <!-- [[Dihedral group|Dih]]('''S'''<sup>1</sup>), where '''S'''<sup>1</sup> denotes the multiplicative group of [[complex number]]s of [[absolute value]] 1. -->
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| It is isomorphic (as a ''real'' Lie group) to the [[circle group]], also known as {{math|[[unitary group|U]](1)}}. This isomorphism sends the complex number {{math|1=exp(φ ''i'') = cos φ + ''i'' sin φ}} of [[absolute value]] 1 to the orthogonal matrix
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| :<math>\begin{bmatrix}\cos(\phi)&-\sin(\phi)\\ \sin(\phi)&\cos(\phi)\end{bmatrix}.</math>
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| The group [[Rotation group SO(3)|{{math|SO(3, '''R''')}}]], understood as the set of rotations of 3-dimensional space, is of major importance in the sciences and engineering, and there are numerous [[charts on SO(3)]].
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| === Low-dimensional topology ===
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| The low dimensional (real) orthogonal groups are familiar [[topological space|space]]s:
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| * {{math|1=O(1) = {±1}}}, a [[2 (number)|two]]-point [[discrete topology|discrete space]]
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| * {{math|1=SO(1) = [[singleton (mathematics)|{1}]]}}
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| * {{math|SO(2)}} is [[circle|{{math|''S''<sup>1</sup>}}]]
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| * [[Rotation group SO(3)|{{math|SO(3)}}]] is [[real projective space|{{math|'''RP'''<sup>3</sup>}}]]
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| * {{math|SO(4)}} is [[double cover (topology)|double covered]] by {{math|1=[[special unitary group|SU]](2) × SU(2) = [[3-sphere|''S''<sup>3</sup>]] × ''S''<sup>3</sup>}}.
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| === Homotopy groups ===
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| In terms of [[algebraic topology]], for {{math|''n'' > 2}} the [[fundamental group]] of {{math|SO(''n'', '''R''')}} is [[Cyclic group|cyclic of order 2]], and the [[spin group]] {{math|Spin(''n'')}} is its [[universal cover]]. For {{math|1=''n'' = 2}} the fundamental group is [[infinite cyclic]] and the universal cover corresponds to the [[real line]] (the group {{math|Spin(2)}} is the unique [[double cover (topology)|2-fold cover]]).
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| Generally, the [[homotopy group]]s {{math|π<sub>''k''</sub>(''O'')}} of the real orthogonal group are related to [[homotopy groups of spheres]], and thus are in general hard to compute. However, one can compute the homotopy groups of the stable orthogonal group (aka the infinite orthogonal group), defined as the [[direct limit]] of the sequence of inclusions:
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| :<math>\mathrm{O}(0) \subset \mathrm{O}(1)\subset \mathrm{O}(2)\subset\cdots\subset O = \bigcup_{k=0}^\infty \mathrm{O}(k)</math>
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| Since the inclusions are all closed, hence [[cofibration]]s, this can also be interpreted as a union. On the other hand [[n-sphere|{{math|''S''<sup>''n''</sup>}}]] is a [[homogeneous space]] for {{math|O(''n'' + 1)}}, and one has the following [[fiber bundle]]:
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| : <math>\mathrm{O}(n) \to \mathrm{O}(n+1) \to S^n,</math>
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| which can be understood as "The orthogonal group {{math|O(''n'' + 1)}} acts [[transitive action|transitively]] on the unit sphere {{math|''S''<sup>''n''</sup>}}, and the [[stabilizer (group theory)|stabilizer]] of a point (thought of as a [[unit vector]]) is the orthogonal group of the [[orthogonal complement|perpendicular complement]], which is an orthogonal group one dimension lower. Thus the natural inclusion {{math|O(''n'') → O(''n'' + 1)}} is [[n-connected|{{math|(''n'' − 1)}}-connected]], so the homotopy groups stabilize, and {{math|1=π<sub>''k''</sub>( π<sub>''k''</sub>(O(''n''))}} for {{math|''n'' > ''k'' + 1}}: thus the homotopy groups of the stable space equal the lower homotopy groups of the unstable spaces.
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| From [[Bott periodicity]] we obtain {{math|Ω<sup>8</sup>''O'' ≅ ''O''}}, therefore the homotopy groups of {{math|''O''}} are 8-fold periodic, meaning {{math|1=π<sub>''k'' + 8</sub>(''O'') = π<sub>''k''</sub>(''O'')}}, and one needs only to compute the lower 8 homotopy groups:
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| : <math>\begin{align}
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| \pi_0 (O) &= \mathbf Z/2\\
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| \pi_1 (O) &= \mathbf Z/2\\
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| \pi_2 (O) &= 0\\
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| \pi_3 (O) &= \mathbf Z\\
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| \pi_4 (O) &= 0\\
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| \pi_5 (O) &= 0\\
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| \pi_6 (O) &= 0\\
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| \pi_7 (O) &= \mathbf Z
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| \end{align}</math>
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| ==== Relation to KO-theory ====
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| Via the [[clutching construction]], homotopy groups of the stable space {{math|''O''}} are identified with stable vector bundles on spheres ([[up to isomorphism]]), with a dimension shift of 1: {{math|1=π<sub>''k''</sub>(''O'') = π<sub>''k'' + 1</sub>(''BO'')}}. Setting {{math|1=''KO'' = ''BO'' × '''Z''' = Ω<sup>−1</sup>''O'' × '''Z'''}} (to make {{math|π<sub>0</sub>}} fit into the periodicity), one obtains:
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| : <math>\begin{align}
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| \pi_0 (KO) &= \mathbf Z\\
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| \pi_1 (KO) &= \mathbf Z/2\\
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| \pi_2 (KO) &= \mathbf Z/2\\
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| \pi_3 (KO) &= 0\\
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| \pi_4 (KO) &= \mathbf Z\\
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| \pi_5 (KO) &= 0\\
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| \pi_6 (KO) &= 0\\
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| \pi_7 (KO) &= 0
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| \end{align}</math>
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| ==== Computation and interpretation of homotopy groups ====
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| ===== Low-dimensional groups =====
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| The first few homotopy groups can be calculated by using the concrete descriptions of low-dimensional groups.
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| *{{math|1=π<sub>0</sub>(''O'') = π<sub>0</sub>(O(1)) = '''Z'''/2}}, from [[orientation (mathematics)|orientation]]-preserving/reversing (this class survives to {{math|O(2)}} and hence stably)
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| *{{math|1=π<sub>1</sub>(''O'') = π<sub>1</sub>(SO(3)) = '''Z'''/2}}, which is [[spin group|spin]] comes from {{math|1=SO(3) = '''RP'''<sup>3</sup> = ''S''<sup>3</sup>/('''Z'''/2)}}.
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| *{{math|1=π<sub>2</sub>(''O'') = π<sub>2</sub>(SO(3)) = 0}}, which surjects onto {{math|π<sub>2</sub>(SO(4))}}; this latter thus vanishes.
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| ===== Lie groups =====
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| From general facts about [[Lie group]]s, {{math|π<sub>2</sub>(''G'')}} always vanishes, and {{math|π<sub>3</sub>(''G'')}} is free ([[free abelian group|free abelian]]).
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| ===== Vector bundles =====
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| From the vector bundle point of view, {{math|π<sub>0</sub>(''K''O)}} is vector bundles over {{math|''S''<sup>0</sup>}}, 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 {{math|1=π<sub>0</sub>(''K''O) = '''[[integers|Z]]'''}} is [[Hamel dimension|dimension]].
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| ===== Loop spaces =====
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| Using concrete descriptions of the loop spaces in [[Bott periodicity]], one can interpret higher homotopy of {{math|''O''}} as lower homotopy of simple to analyze spaces. Using π<sub>0</sub>, {{math|''O''}} and {{math|''O''/U}} have two components, {{math|1=''K''O = ''B''O × '''Z'''}} and {{math|1=''K''Sp = ''B''Sp × '''Z'''}} have [[countably many]] components, and the rest are connected.
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| ==== Interpretation of homotopy groups ====
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| In a nutshell:<ref>[http://math.ucr.edu/home/baez/week105.html John Baez "This Week's Finds in Mathematical Physics" week 105]</ref>
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| *{{math|1=π<sub>0</sub>(''K''O) = '''Z'''}} is about [[Hamel dimension|dimension]]
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| *{{math|1=π<sub>1</sub>(''K''O) = '''Z'''/2}} is about [[orientation (mathematics)|orientation]]
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| *{{math|1=π<sub>2</sub>(''K''O) = '''Z'''/2}} is about [[spin group|spin]]
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| *{{math|1=π<sub>4</sub>(''K''O) = '''Z'''}} is about [[topological quantum field theory]].
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| Let {{mvar|R}} be any of four [[division ring]]s {{math|'''R''', '''C''', '''[[quaternions|H]]''', '''[[octonions|O]]'''}},<!-- since '''H''' and '''O''' are not fields, and also to avoid conflict with the lead, I change the former ''F'' notation to ''R'' (ring) --Incnis Mrsi --> and let ''L<sub>R</sub>'' be the [[tautological line bundle]] over the [[projective line]] {{math|''R'''''P'''<sup>1</sup>}}, and {{math|[''L<sub>R</sub>'']}} its class in K-theory. Noting that {{math|1=[[real projective line|'''RP'''<sup>1</sup>]] = ''S''<sup>1</sup>}}, {{math|1=[[Riemann sphere|'''CP'''<sup>1</sup>]] = ''S''<sup>2</sup>}}, {{math|1='''HP'''<sup>1</sup> = ''S''<sup>4</sup>}}, {{math|1='''OP'''<sup>1</sup> = ''S''<sup>8</sup>}}, these yield vector bundles over the corresponding spheres, and
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| *{{math|π<sub>1</sub>(''K''O)}} is generated by {{math|[''L''<sub>'''R'''</sub>]}}
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| *{{math|π<sub>2</sub>(''K''O)}} is generated by {{math|[''L''<sub>'''C'''</sub>]}}
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| *{{math|π<sub>4</sub>(''K''O)}} is generated by {{math|[''L''<sub>'''H'''</sub>]}}
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| *{{math|π<sub>8</sub>(''K''O)}} is generated by {{math|[''L''<sub>'''O'''</sub>]}}
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| From the point of view of [[symplectic geometry]], {{math|1=π<sub>0</sub>(''K''O) ≅ π<sub>8</sub>(''K''O) = '''Z'''}} can be interpreted as the [[Maslov index]], thinking of it as the fundamental group {{math|π<sub>1</sub>(U/O)}} of the stable [[Lagrangian Grassmannian]] as {{math|U/O ≅ Ω<sup>7</sup>(''K''O)}}, so {{math|1=π<sub>1</sub>(U/O) = π<sub>1 + 7</sub>(''K''O)}}.
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| == Over the complex number field ==
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| Over the field {{math|'''C'''}} of [[complex number]]s, {{math|O(''n'', '''C''')}} and {{math|SO(''n'', '''C''')}} are complex Lie groups of dimension {{math|''n''(''n'' − 1)/2}} over '''C''' (it means the dimension over {{math|'''R'''}} is twice that). {{math|O(''n'', '''C''')}} has two connected components, and {{math|SO(''n'', '''C''')}} is the connected component containing the identity matrix. For {{math|''n'' ≥ 2}} these groups are noncompact.
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| Just as in the real case {{math|SO(''n'', '''C''')}} is not simply connected. For {{math|''n'' > 2}} the [[fundamental group]] of {{math|SO(''n'', '''C''')}} is [[Cyclic group|cyclic of order 2]] whereas the fundamental group of {{math|SO(2, '''C''')}} is [[infinite cyclic]].
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| ==Over finite fields==
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| Orthogonal groups can also be defined over [[finite field]]s {{math|'''F'''<sub>''q''</sub>}}, where {{mvar|q}} is a power of a prime {{mvar|p}}. When defined over such fields, they come in two types{{clarify|date=February 2013}} in even dimension: {{math|O<sup>+</sup>(2''n'', ''q'')}} and {{math|O<sup>−</sup>(2''n'', ''q'')}}; and one type in odd dimension: {{math|O(2''n'' + 1, ''q'')}}.
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| If {{mvar|V}} is the vector space on which the orthogonal group {{mvar|G}} acts, it can be written as a direct orthogonal sum as follows:
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| : <math> V = L_1 \oplus L_2 \oplus \cdots \oplus L_m \oplus W, </math>
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| where {{math|''L<sub>i</sub>''}} are [[hyperbolic line]]s and {{mvar|W}} contains no singular vectors. If {{mvar|W}} is the [[zero vector space|zero subspace]], then {{mvar|G}} is of plus type. If {{mvar|W}} is one-dimensional then {{mvar|G}} has odd dimension. If {{mvar|W}} has dimension 2, {{mvar|G}} is of minus type.
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| In the special case where {{math|1=''n'' = 1}}, {{math|O<sup>''ϵ''</sup>(2, ''q'')}} is a [[dihedral group]] of order {{math|2(''q'' − ''ϵ'')}}.
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| We have the following formulas for the order of {{math|O(''n'', ''q'')}}, when the characteristic is greater than two:
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| : <math>|\mathrm{O}(2n+1,q)|=2q^n\prod_{i=0}^{n-1}(q^{2n}-q^{2i}).</math>
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| If −1 is a [[quadratic residue<!-- formally, only if q = p. does a better link, but square (algebra), exist? -->|square in {{math|'''F'''<sub>''q''</sub>}}]]
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| : <math>|\mathrm{O}(2n,q)|=2(q^n-1)\prod_{i=1}^{n-1}(q^{2n}-q^{2i}).</math>
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| If −1 is a non-square in {{math|'''F'''<sub>''q''</sub>}}
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| :<math>|\mathrm{O}(2n,q)|=2(q^n+(-1)^{n+1})\prod_{i=1}^{n-1}(q^{2n}-q^{2i}).</math>
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| == The Dickson invariant ==
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| For orthogonal groups, the '''Dickson invariant''' is a homomorphism from the orthogonal group to the quotient group {{math|'''Z'''/2'''Z'''}} (integers modulo 2), taking the value 0 in case the element is the product of an even number of reflections, and the value of 1 otherwise.<ref name=Knus224>{{citation | last=Knus | first=Max-Albert | title=Quadratic and Hermitian forms over rings | series=Grundlehren der Mathematischen Wissenschaften | volume=294 | location=Berlin etc. | publisher=[[Springer-Verlag]] | year=1991 | isbn=3-540-52117-8 | zbl=0756.11008 | page=224 }}</ref>
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| Algebraically, the Dickson invariant can be defined as {{math|1=''D''(''f'') = rank(''I'' − ''f'') modulo 2}}, where {{math|''I''}} is the identity {{harv|Taylor|1992|loc=Theorem 11.43}}. Over fields that are not of [[characteristic (algebra)|characteristic]] 2 it is equivalent to the determinant: the determinant is −1 to the power of the Dickson invariant.
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| Over fields of characteristic 2, the determinant is always 1, so the Dickson invariant gives more information than the determinant.
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| The special orthogonal group is the [[Kernel (matrix)|kernel]] of the Dickson invariant<ref name=Knus224/> and usually has index 2 in {{math|O(''n'', ''F'' )}}.<ref>{{harv|Taylor|1992|loc=page 160}}</ref> 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, {{math|SO(''n'', ''F'' )}} is commonly defined to be the elements of {{math|O(''n'', ''F'' )}} with determinant 1. Each element in {{math|O(''n'', ''F'' )}} has determinant ±1. Thus in characteristic 2, the determinant is always 1.
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| The Dickson invariant can also be defined for [[Clifford group]]s and [[Pin group]]s in a similar way (in all dimensions).
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| ==Orthogonal groups of characteristic 2==
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| 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.)
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| *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.<ref>{{harv|Grove|2002|loc=Theorem 6.6 and 14.16}}</ref> Note that a reflection in characteristic two has a slightly different definition. In characteristic two, the reflection orthogonal to a vector {{math|'''u'''}} takes a vector {{math|'''v'''}} to {{math|'''v''' + ''B''('''v''', '''u''')/Q('''u''') · '''u'''}} where {{math|''B''}} is the bilinear form and {{math|''Q''}} is the quadratic form associated to the orthogonal geometry. Compare this to the [[Householder reflection]] of odd characteristic or characteristic zero, which takes {{math|'''v'''}} to {{math|'''v''' − 2·''B''('''v''', '''u''')/Q('''u''') · '''u'''}}.
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| *The [[center of a group|center]] of the orthogonal group usually has order 1 in characteristic 2, rather than 2, since {{math|1=''I'' = −''I''}}.
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| *In odd dimensions {{math|2''n'' + 1}} in characteristic 2, orthogonal groups over [[perfect field]]s are the same as [[symplectic group]]s in dimension {{math|2''n''}}. 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 {{math|2''n''}}, acted upon by the orthogonal group.
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| *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.
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| == The spinor norm ==<!-- This section is linked from [[Clifford algebra]] -->
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| The '''spinor norm''' is a homomorphism from an orthogonal group over a field {{mvar|F}} to the [[quotient group]] {{math|''F'' */''F'' *<sup>2</sup>}} (the [[multiplicative group]] of the field {{mvar|F}} up to [[square (algebra)|square]] elements), that takes reflection in a vector of norm {{mvar|n}} to the image of {{mvar|n}} in {{math|''F'' */''F'' *<sup>2</sup>}}.<ref name=C178>{{harvnb|Cassels|1978|p=178}}</ref>
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| 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.
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| ==Galois cohomology and orthogonal groups==
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| In the theory of [[Galois cohomology]] of [[algebraic group]]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]]s over a field can be identified as a Galois {{math|''H''<sup>1</sup>}}, or twisted forms ([[torsor]]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 [[discriminant]].
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| 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 algebra]]s). The spin covering of the orthogonal group provides a [[short exact sequence]] of [[algebraic group]]s.
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| :<math> 1 \rightarrow \mu_2 \rightarrow \mathrm{Pin}_V \rightarrow \mathrm{O_V} \rightarrow 1 </math>
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| Here {{math|μ<sub>2</sub>}} is the [[Group scheme of roots of unity|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 {{math|''H''<sup>0</sup>(O<sub>V</sub>)}}, which is simply the group {{math|O<sub>V</sub>(''F'' )}} of {{mvar|F}}-valued points, to {{math|''H''<sup>1</sup>(μ<sub>2</sub>)}} is essentially the spinor norm, because {{math|''H''<sup>1</sup>(μ<sub>2</sub>)}} is isomorphic to the multiplicative group of the field modulo squares.
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| There is also the connecting homomorphism from {{math|''H''<sup>1</sup>}} of the orthogonal group, to the {{math|''H''<sup>2</sup>}} 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.
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| == Lie algebra ==
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| {{anchor|orthogonal Lie algebra}}{{anchor|special orthogonal Lie algebra}}
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| The [[Lie algebra]] corresponding to Lie groups {{math|O(''n'', ''F'' )}} and {{math|SO(''n'', ''F'' )}} consists of the [[skew-symmetric matrix|skew-symmetric]] {{math|''n'' × ''n''}} matrices, with the Lie bracket {{math|[ , ]}} given by the [[commutator]]. One Lie algebra corresponds to both groups. It is often denoted by <math>\mathfrak{o}(n, F)</math> or <math>\mathfrak{so}(n, F)</math>, and called the '''orthogonal Lie algebra''' or '''special orthogonal Lie algebra'''. Over real numbers, these Lie algebras for different {{mvar|n}} are the [[compact real form]]s of two of the four families of [[semisimple Lie algebra]]s: in odd dimension {{math|B<sub>''k''</sub>}}, where {{math|1=''n'' = 2''k'' + 1}}, while in even dimension {{math|D<sub>''r''</sub>}}, where {{math|1=''n'' = 2''r''}}.
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| More intrinsically, given a vector space with an inner product, the special orthogonal Lie algebra is given by the [[bivector]]s on the space, which are sums of simple bivectors ([[blade (geometry)|2-blades]]) {{math|'''v''' ∧ '''w'''}}. The correspondence is given by the map <math>\mathbf{v} \wedge \mathbf{w} \mapsto \mathbf{v}^* \otimes \mathbf{w} - \mathbf{w}^* \otimes \mathbf{v},</math> where {{math|'''v'''*}} is the [[covector]] dual to the vector {{math|'''v'''}}; in coordinates these are exactly the elementary skew-symmetric matrices.
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| Over real numbers, this characterization is used in interpreting the [[curl (mathematics)|curl]] of a vector field (naturally a 2-vector) as an infinitesimal rotation or "curl", hence the name. Generalizing the inner product with a [[nondegenerate form]] yields the indefinite orthogonal Lie algebras <math>\mathfrak{so}(p, q).</math>
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| The representation theory of the orthogonal Lie algebras includes both representations corresponding to linear representations of the orthogonal groups, and representations corresponding to projective representations of the orthogonal groups (linear representations of spin groups), the so-called [[spin representation]], which are important in physics.
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| ==Related groups==
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| The orthogonal groups and special orthogonal groups have a number of important subgroups, supergroups, quotient groups, and covering groups. These are listed below.
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| The inclusions {{math|1=O(''n'') ⊂ U(''n'') ⊂ Sp(''n'') = USp(2''n'')}} and {{math|1=USp(''n'') ⊂ U(''n'') ⊂ O(2''n'')}} are part of a sequence of 8 inclusions used in a [[Bott periodicity theorem#Geometric model of loop spaces|geometric proof of the Bott periodicity theorem]], and the corresponding quotient spaces are [[symmetric space]]s of independent interest – for example, {{math|U(''n'')/O(''n'')}} is the [[Lagrangian Grassmannian]].
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| ===Lie subgroups===
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| 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:
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| :<math> \mathrm{O}(n) \supset \mathrm{O}(n-1) </math> – preserve an axis
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| :<math> \mathrm{O}(2n) \supset \mathrm{U}(n) \supset \mathrm{SU}(n) </math> – {{math|U(''n'')}} are those that preserve a compatible complex structure ''or'' a compatible symplectic structure – see [[Unitary group#2-out-of-3 property|2-out-of-3 property]]; {{math|SU(''n'')}} also preserves a complex orientation.
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| :<math> \mathrm{O}(2n) \supset \mathrm{USp}(n) </math>
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| :<math> \mathrm{O}(7) \supset \mathbf{G}_2 </math>
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| ===Lie supergroups===
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| The orthogonal group {{math|O(''n'')}} is also an important subgroup of various Lie groups:
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| :<math> \mathrm{U}(n) \supset \mathrm{SU}(n) \supset \mathrm{O}(n) </math>
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| :<math> \mathrm{USp}(2n) \supset \mathrm{O}(n) </math>
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| :<math> \mathbf{G}_2 \supset \mathrm{O}(3) </math>
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| :<math> \mathbf{F}_4 \supset \mathrm{O}(9) </math>
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| :<math> \mathbf{E}_6 \supset \mathrm{O}(10) </math>
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| :<math> \mathbf{E}_7 \supset \mathrm{O}(12) </math>
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| :<math> \mathbf{E}_8 \supset \mathrm{O}(16) </math>
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| ====Conformal group====
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| {{Main|Conformal group}}
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| Being [[isometry|isometries]], real orthogonal transforms preserve [[angle]]s, and are thus [[conformal map]]s, though not all conformal linear transforms are orthogonal. In classical terms this is the difference between [[congruence (geometry)|congruence]] and [[similarity (geometry)|similarity]], as exemplified by SSS (Side-Side-Side) [[Congruence (geometry)#Congruence of triangles|congruence of triangles]] and AAA (Angle-Angle-Angle) [[similar triangles|similarity of triangles]]. The group of conformal linear maps of {{math|'''R'''<sup>''n''</sup>}} is denoted {{math|CO(''n'')}} for the '''conformal orthogonal group''', and consists of the product of the orthogonal group with the group of [[homothety|dilations]]. If {{mvar|n}} is odd, these two subgroups do not intersect, and they are a [[direct product of groups|direct product]]: {{math|1=CO(2''k'' + 1) = O(2''k'' + 1) × '''R'''*}}, where {{math|1='''R'''* = '''R'''\{0}}} is the real [[multiplicative group]], while if {{mvar|n}} is even, these subgroups intersect in ±1, so this is not a direct product, but it is a direct product with the subgroup of dilation by a positive scalar: {{math|1=CO(2''k'') = O(2''k'') × '''R'''<sup>+</sup>}}.
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| Similarly one can define {{math|CSO(''n'')}}; note that this is always: {{math|1=CSO(''n'') = CO(''n'') ∩ GL<sup>+</sup>(''n'') = SO(''n'') × '''R'''<sup>+</sup>}}.
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| ===Discrete subgroups===
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| As the orthogonal group is compact, discrete subgroups are equivalent to finite subgroups.<ref group="note">Infinite subsets of a compact space have an [[accumulation point]] and are not discrete.</ref> These subgroups are known as [[point group]] and can be realized as the symmetry groups of [[polytope]]s. A very important class of examples are the [[finite Coxeter group]]s, which include the symmetry groups of [[regular polytope]]s.
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| Dimension 3 is particularly studied – see [[point groups in three dimensions]], [[polyhedral group]]s, and [[list of spherical symmetry groups]]. In 2 dimensions, the finite groups are either cyclic or dihedral – see [[point groups in two dimensions]].
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| Other finite subgroups include:
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| * [[Permutation matrices]] (the [[Coxeter group]] {{math|A<sub>''n''</sub>}})
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| * [[Signed permutation matrices]] (the [[Coxeter group]] {{math|B<sub>''n''</sub>}}); also equals the intersection of the orthogonal group with the [[integer matrices]].<ref group="note">{{math|O(''n'') ∩ [[general linear group|GL]](''n'', '''Z''')}} equals the signed permutation matrices because an integer vector of norm 1 must have a single non-zero entry, which must be ±1 (if it has two non-zero entries or a larger entry, the norm will be larger than 1), and in an orthogonal matrix these entries must be in different coordinates, which is exactly the signed permutation matrices.</ref>
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| ===Covering and quotient groups===
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| The orthogonal group is neither [[simply connected]] nor [[centerless]], and thus has both a [[covering group]] and a [[quotient group]], respectively:
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| * Two covering [[Pin group]]s, {{math|Pin<sub>+</sub>(''n'') → O(''n'')}} and {{math|Pin<sub>−</sub>(''n'') → O(''n'')}},
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| * The quotient [[projective orthogonal group]], {{math|O(''n'') → PO(''n'')}}.
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| These are all 2-to-1 covers.
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| For the special orthogonal group, the corresponding groups are:
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| * [[Spin group]], {{math|Spin(''n'') → SO(''n'')}},
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| * [[Projective special orthogonal group]], {{math|SO(''n'') → PSO(''n'')}}.
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| Spin is a 2-to-1 cover, while in even dimension, {{math|PSO(2''k'')}} is a 2-to-1 cover, and in odd dimension {{math|PSO(2''k'' + 1)}} is a 1-to-1 cover, i.e., isomorphic to {{math|SO(2''k'' + 1)}}. These groups, {{math|Spin(''n'')}}, {{math|SO(''n'')}}, and {{math|PSO(''n'')}} are Lie group forms of the compact [[special orthogonal Lie algebra]], <math>\mathfrak{so}(n, {\mathbb R})</math> – Spin is the simply connected form, while PSO is the centerless form, and SO is in general neither.<ref group="note">In odd dimension, {{math|SO(2''k'' + 1) ≅ PSO(2''k'' + 1)}} is centerless (but not simply connected), while in even dimension {{math|SO(2''k'')}} is neither centerless nor simply connected.</ref>
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| In dimension 3 and above these are the covers and quotients, while dimension 2 and below are somewhat degenerate; see specific articles for details.
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| ==Principal homogeneous space: Stiefel manifold==
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| {{Main|Stiefel manifold}}
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| The [[principal homogeneous space]] for the orthogonal group {{math|O(''n'')}} is the [[Stiefel manifold]] {{math|''V<sub>n</sub>''('''R'''<sup>''n''</sup>)}} of [[orthonormal bases]] (orthonormal [[k-frame|{{mvar|n}}-frames]]).
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| 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.
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| The other Stiefel manifolds {{math|''V<sub>k</sub>''('''R'''<sup>''n''</sup>)}} for {{math|''k'' < ''n''}} of ''incomplete'' orthonormal bases (orthonormal {{mvar|k}}-frames) are still homogeneous spaces for the orthogonal group, but not ''principal'' homogeneous spaces: any {{mvar|k}}-frame can be taken to any other {{mvar|k}}-frame by an orthogonal map, but this map is not uniquely determined.
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| ==See also==
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| ===Specific transforms===
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| *[[Coordinate rotations and reflections]]
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| *[[Reflection through the origin]]
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| ===Specific groups===
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| *rotation group, [[Rotation group SO(3)|SO(3, '''R''')]]
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| *[[SO(8)]]
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| ===Related groups===
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| *[[indefinite orthogonal group]]
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| *[[unitary group]]
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| *[[symplectic group]]
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| ===Lists of groups===
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| *[[list of finite simple groups]]
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| *[[list of simple Lie groups]]
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| ==Notes==
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| {{Reflist| group = note }}
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| ==References==
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| {{Reflist}}
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| {{Refimprove|date=May 2010}}
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| {{refbegin}}
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| * {{citation | first=J.W.S. | last=Cassels | authorlink=J. W. S. Cassels | title=Rational Quadratic Forms | series=London Mathematical Society Monographs | volume=13 | publisher=[[Academic Press]] | year=1978 | isbn=0-12-163260-1 | zbl=0395.10029 }}
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| * {{Citation | last1=Grove | first1=Larry C. | title=Classical groups and geometric algebra | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Graduate Studies in Mathematics | isbn=978-0-8218-2019-3 | mr=1859189 | year=2002 | volume=39}}
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| * {{Citation | last1=Taylor | first1=Donald E. | title=The Geometry of the Classical Groups | publisher=Heldermann Verlag | location=Berlin | isbn=3-88538-009-9| mr=1189139 | year=1992 | series=Sigma Series in Pure Mathematics | volume=9 | zbl=0767.20001 }}
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| {{refend}}
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| ==External links==
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| *{{springer|title=Orthogonal group|id=p/o070300}}
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| *[http://math.ucr.edu/home/baez/week105.html John Baez "This Week's Finds in Mathematical Physics" week 105]
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| *[http://math.ucr.edu/home/baez/octonions/node10.html John Baez on Octonions]
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| * {{it}} [http://ansi.altervista.org n-dimensional Special Orthogonal Group parametrization]
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| [[Category:Lie groups]]
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| [[Category:Quadratic forms]]
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| [[Category:Euclidean symmetries]]
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