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{{short description|Group of isometries of a Euclidean vector space or, more generally, of a vector space equipped with a quadratic form}}

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In [[mathematics]], the '''orthogonal group''' in dimension {{math|''n''}} (sometimes called '''general orthogonal group''', by analogy with the [[general linear group]]) , denoted {{math|O(''n'')}}, is the [[Group (mathematics)|group]] of [[isometry|distance-preserving transformations]] of a [[Euclidean space]] of dimension {{math|''n''}} that preserve a fixed point, where the group operation is given by [[Function composition|composing]] transformations. Equivalently, it is the group of {{math|''n''×''n''}} [[orthogonal matrix|orthogonal matrices]], 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 orthogonal group is an [[algebraic group]] and a [[Lie group]]. It is [[compact group|compact]].

The orthogonal group in dimension {{math|''n''}} has two [[connected component (topology)|connected component]]s. The one that contains the [[identity element]] is a subgroup, called the '''special orthogonal group''', and denoted {{math|SO(''n'')}}. It consists of all orthogonal matrices of [[determinant]] {{math|1}}. In the other connected component all orthogonal matrices have {{math|–1}} as a determinant. This group is also called the '''rotation group''', because, in dimensions 2 and 3, its elements are the usual [[rotation (mathematics)|rotations]] around a point (in dimension 2) or a line (in dimension 3). In low dimension, these groups have been widely studied, see {{math|[[SO(2)]]}}, {{math|[[SO(3)]]}} and {{math|[[SO(4)]]}}.

By extension, for any field {{mvar|F}}, a {{math|''n''×''n''}} matrix with entries in {{mvar|F}} such that its inverse equals its transpose is called an ''orthogonal matrix over'' {{mvar|F}}. The {{math|''n''×''n''}} orthogonal
matrices form a subgroup, denoted {{math|O(''n'', ''F'')}}, of the [[general linear group]] {{math|GL(''n'', ''F'')}}; that is 
:<math>\operatorname{O}(n, F) = \left\{Q \in \operatorname{GL}(n, F) \mid Q^\mathsf{T} Q = Q Q^\mathsf{T} = I \right\} .</math>

More generally, given a non-degenerate [[symmetric bilinear form]] or [[quadratic form]]<ref>For base fields of [[Characteristic (algebra)|characteristic]] not 2, the definition in terms of a [[symmetric bilinear form]] is equivalent to that in terms of a [[quadratic form]], but in characteristic 2 these notions differ.</ref> on a [[vector space]] over a [[field (mathematics)|field]], the ''orthogonal group of the form'' is the group of invertible [[linear map]]s that preserve the form. The preceding orthogonal groups are the spacial case where, on some basis, the bilinear form is the [[dot product]], or, equivalently, the quadratic form is the sum of the square of the coordinates.

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

==Name==

The name of "orthogonal group" originates in the following characterization of its elements. Given a [[Euclidean vector space]] {{mvar|E}} of dimension {{mvar|n}}, the elements of the orthogonal group {{math|O(''n'')}} are the [[linear map]]s  from {{mvar|E}} to {{mvar|E}} that map [[orthogonal vector]]s to orthogonal vectors.

==In Euclidean geometry==

The orthogonal group {{math|O(''n'')}} is the subgroup of the [[general linear group]] {{math|GL(''n'', '''R''')}}, consisting of all [[endomorphisms]] that preserve the [[Euclidean norm]], that is endomorphisms {{mvar|g}} such that <math>\|g(x)\| = \|x\|.</math>

Let {{math|E(''n'')}} be the group of the [[Euclidean isometry|Euclidean isometries]] of a [[Euclidean space]] {{mvar|S}} of dimension {{mvar|n}}. This group does not depend on the choice of a particular space, since all Euclidean spaces of the same dimension are [[isomorphic]]. The [[stabilizer subgroup]] of a point {{math|''x'' ∈ ''S''}} is the subgroup of the elements {{math|''g'' ∈ E(''n'')}} such that {{math|1=''g''(''x'') = ''x''}}. This stabilizer is (or, more exactly, is isomorphic to) {{math|O(''n'')}}, since the choice of a point as an origin induces an isomorphism between the Euclidean space and its associated Euclidean vector space.

There is a natural [[group homomorphism]] {{mvar|p}} from {{math|E(''n'')}} to {{math|O(''n'')}}, which is defined by
:<math>p(g)\cdot (y-x) = g(y)-g(x),</math>
where, as usual, the subtraction of two points denotes the [[translation (geometry)|translation]] vector that maps the second point to the first one. This is a well defined homomorphism, since a straightforward verification shows that, if two pairs of points have the same difference, the same is true for their images by {{mvar|g}} (for details, see {{slink|Affine space|Subtraction and Weyl's axioms}}).

The [[kernel (algebra)|kernel]] of {{mvar|p}} is the vector space of the translations. So, the translation form a [[normal subgroup]] of {{math|E(''n'')}}, the stabilizers of two points are [[conjugate subgroup|conjugate]] under te action of the translations, and all stabilizers are isomorphic to {{math|O(''n'')}}. 

Moreover, the Euclidean group is a [[semidirect product]] of {{math|O(''n'')}} and the group of translations. It follows that the study of the Euclidean group is essentially reduced to the study of {{math|O(''n'')}}.

==={{math|SO(''n'')}}===
By choosing an [[orthonormal basis]] of an Euclidean vector space, the orthogonal group can be identified with the group (under matrix multiplication) of [[orthogonal matrices]], which are the matrices such that 
:<math> Q Q^\mathsf{T} = I. </math>

It follows from this equation that the square of the [[determinant]] of {{mvar|Q}} equals {{math|1}}, and thus that the determinant of {{mvar|Q}} is either {{math|1}} or {{math|–1}}. The orthogonal matrices with determinant {{math|1}} form a subgroup of {{math|SO(''n'')}}, called the ''special orthogonal group'', and denoted {{math|SO(''n'')}}. It consists of all [[Euclidean group#Direct and indirect isometries|direct isometries]] of {{math|O(''n'')}}, which are those that preserve the [[orientation (vector space)|orientation]] of the space.

{{math|SO(''n'')}} is a normal subgroup of {{math|O(''n'')}}, as being the [[kernel (algebra)|kernel]] of the determinant, which is here a group homomorphism whose image is the multiplicative group {{math|{–1, +1}.}} Moreover, the orthogonal group is a [[semidirect product]] of {{math|SO(''n'')}} and the group with two elements, since, given any [[reflection (geometry)|reflection]] {{mvar|r}}, one has {{math|1=O(''n'') \ SO(''n'') = ''r'' SO(''n'')}}.

The group with two emements {{math|{±''I''}}} (where {{mvar|I}} is the identity matrix) is a [[normal subgroup]] and even a [[characteristic subgroup]] of {{math|O(''n'')}}, and, if {{math|''n''}} is even, also of {{math|SO(''n'')}}. If {{math|''n''}} is odd, {{math|O(''n'')}} is the internal [[direct product of groups|direct product]] of {{math|SO(''n'')}} and {{math|{±''I''}}}. For every positive integer {{math|''k''}} the [[cyclic group]] {{math|''C''<sub>''k''</sub>}} of [[rotational symmetry|{{math|''k''}}-fold rotations]] is a normal subgroup of {{math|O(2)}} and {{math|SO(2)}}.

===Canonical form===
For any element of {{math|O(''n'')}} there is an orthogonal basis, where its matrix has the form
:<math>\begin{bmatrix}
  \begin{matrix}
    R_1 &        & \\
        & \ddots & \\
        &        & R_k
  \end{matrix} & 0 \\
  0 & \begin{matrix}
    \pm 1 &        & \\
          & \ddots & \\
          &        & \pm 1
  \end{matrix}\\
\end{bmatrix},</math>
where the matrices {{math|''R''<sub>1</sub>, ..., ''R''<sub>''k''</sub>}} are 2-by-2 rotation matrices, that is matrices of the form 
:<math>\begin{bmatrix}a&b\\-b&a\end{bmatrix},</math>
with <math>a^2+b^2=1.</math>

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

The element belongs to {{math|SO(''n'')}} if and only if thre are an even number of {{math|–1}} on the diagona

The special case of {{math|1=''n'' = 3}} is known as [[Euler's rotation theorem]], which asserts that every (non-identity) element of {{math|SO(3)}} is a [[rotation]] about a uniquely defined axis.

===Reflections===
[[Reflection (mathematics)|Reflection]]s are the elements of {{math|O(''n'')}} whose canonical form is 
:<math>\begin{bmatrix}-1&0\\0&I\end{bmatrix},</math>
where {{mvar|I}} is the {{math|(''n''–1)×(''n''–1)}} identity matrix, and the zeros denote row or column zero matrices. In other words, a reflection is a transformation that transforms the space in its [[mirror image]] with respect to a [[hyperplane]].

In dimension two, [[Rotations and reflections in two dimensions|every rotation is the product of two reflections]]. More precisely, a rotation of angle {{mvar|{{theta}}}} is the product of two reflections whose axes have an angle of {{math|''{{theta}}'' / 2}}.

Every element of {{math|O(''n'')}} is the product of at most {{mvar|n}} reflections. This results immediately from the above canonical form and the case of dimension two.

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

The [[reflection through the origin]] (the map {{math|''v'' ↦ −''v''}}) is an example of an element of {{math|O(''n'')}} that is not the product of less than {{mvar|n}} reflections.

===Symmetry group of spheres===
The orthogonal group {{math|O(''n'')}} is the [[symmetry group]] of the [[n-sphere|{{math|(''n'' − 1)}}-sphere]] (for {{math|1=''n'' = 3}}, this is just the [[sphere]]) and all objects with spherical symmetry, if the origin is chosen at the center.

The [[symmetry group]] of a [[circle]] is {{math|O(2)}}.<!--
[[Dihedral group|Dih]]('''S'''<sup>1</sup>), where '''S'''<sup>1</sup> denotes the multiplicative group of [[complex number]]s of [[absolute value]]&nbsp;1.
--> The orientation-preserving subgroup {{math|SO(2)}}  is isomorphic (as a ''real'' Lie group) to the [[circle group]], also known as {{math|[[unitary group|U]](1)}}, the multiplicative group of the [[complex number]]s of absolute value equal to one. This isomorphism sends the complex number {{math|1=exp(''φ'' ''i'') = cos(''φ'') + ''i'' sin(''φ'')}} of [[absolute value]]&nbsp;{{math|1}} to the special orthogonal matrix
:<math>\begin{bmatrix}
  \cos(\varphi) & -\sin(\varphi) \\
  \sin(\varphi) &  \cos(\varphi)
\end{bmatrix}.</math>

In higher dimension, {{math|O(''n'')}} has a more complicate structure (in particular, it is no more commutative). The [[topological]] structures of the {{mvar|n}}-sphere and {{math|O(''n'')}} are strongly correlated, and this correlation is widely used for studying both [[topological space]]s.

==Group structure==
The groups {{math|O(''n'')}} and {{math|SO(''n'')}} are real [[Compact space|compact]] [[Lie group]]s of [[dimension (mathematics)|dimension]] {{math|''n''(''n'' − 1)/2}}. The group {{math|O(''n'')}} has two [[Connected space|connected components]], with {{math|SO(''n'')}} being the [[identity component]], that is, the connected component containing the [[identity matrix]].

===As algebraic groups===
The orthogonal group {{math|O(''n'')}} can be identified with the group of the matrices {{mvar|A}} such that  <math>A^\mathsf{T}A = I.</math> 
Since both members of this equation are [[symmetric matrices]], this provides <math>\textstyle \frac {n(n + 1)}2</math> 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 {{math|O(''n'')}} is an [[algebraic set]]. Moreover, it can be proved that its dimension is
:<math>\frac{n(n - 1)}{2} = n^2 - \frac{n(n + 1)}{2},</math> 
which implies that {{math|O(''n'')}} is a [[complete intersection]]. This implies that all its [[irreducible component]]s have the same dimension, and that it has no [[embedded prime|embedded component]].
In fact, {{math|O(''n'')}} has two irreducible components, that are distinguished by the sign of the determinant (that is {{math|1=det(''A'') = 1}} or {{math|1=det(''A'') = –1}}). Both are [[singular point of an algebraic variety|nonsingular algebraic varieties]] of the same dimension {{math|''n''(''n'' – 1) / 2}}. The component with {{math|1=det(''A'') = 1}} is {{math|SO(''n'')}}.

===Maximal tori and Weyl groups===
A [[maximal torus]] in a compact [[Lie group]] ''G'' is a maximal subgroup among those that are isomorphic to {{math|'''T'''<sup>''k''</sup>}} for some {{mvar|k}}, where {{math|1='''T''' = SO(2)}} is the standard one-dimensional torus.<ref>{{harvnb|Hall|2015}} Theorem 11.2</ref>

In {{math|O(2''n'')}} and {{math|SO(2''n'')}}, for every maximal torus, there is a basis on which the torus consists of the [[block matrix#Block diagonal matrices|block-diagonal matrices]] of the form 
:<math>\begin{bmatrix}
  R_1 &        & 0    \\
      & \ddots &      \\
  0   &        & R_n
\end{bmatrix},</math>
where each {{math|''R''<sub>''j''</sub>}} belongs to {{math|SO(2)}}. 
In {{math|O(2''n'' + 1)}} and {{math|SO(2''n'' + 1)}}, the maximal tori have the same form, bordered by a row and a column of zeros, and 1 on the diagonal.

The [[Maximal torus#Weyl group|Weyl group]]<!--Do not change the link (unless the target atricle(s) have changed: Presently, the article [[Weyl group]] does not talk of the Weyl group relatively to a maximal torus) --> of {{math|SO(2''n'' + 1)}} is the [[semidirect product]] <math>\{\pm 1\}^n \rtimes S_n</math> of a normal [[elementary abelian group|elementary abelian]] [[p-group|2-subgroup]] and a [[symmetric group]], where the nontrivial element of each {{math|{±1}}} factor of {{math|{±1}<sup>''n''</sup>}} acts on the corresponding circle factor of {{math|''T'' × {1}}} by [[multiplicative inverse|inversion]], and the symmetric group {{math|''S<sub>n</sub>''}} acts on both {{math|{±1}<sup>''n''</sup>}} and {{math|''T'' × {1}}} by permuting factors. The elements of the Weyl group are represented by matrices in {{math|O(2''n'') × {±1}}}.
The {{math|''S<sub>n</sub>''}} factor is represented by block permutation matrices with 2-by-2 blocks, and a final 1 on the diagonal. The {{math|{±1}<sup>''n''</sup>}} component is represented by block-diagonal matrices with 2-by-2 blocks either
:<math>\begin{bmatrix}
    1 & 0 \\
    0 & 1
  \end{bmatrix} \quad \text{or} \quad
  \begin{bmatrix}
    0 & 1 \\
    1 & 0
  \end{bmatrix},
</math>

with the last component {{math|±1}} chosen to make the determinant 1.

The Weyl group of {{math|SO(2''n'')}} is the subgroup <math>H_{n-1} \rtimes S_n < \{\pm 1\}^n \rtimes S_n</math> of that of {{math|SO(2''n'' + 1)}}, where {{math|''H''<sub>''n''−1</sub> < {±1}<sup>''n''</sup>}} is the [[kernel (algebra)#Group homomorphism|kernel]] of the product homomorphism {{math|{±1}<sup>''n''</sup> → {±1}}} given by <math>\left(\epsilon_1, \ldots, \epsilon_n\right) \mapsto \epsilon_1 \cdots \epsilon_n</math>; that is, {{math|''H''<sub>''n''−1</sub> < {±1}<sup>''n''</sup>}} is the subgroup with an even number of minus signs. The Weyl group of {{math|SO(2''n'')}} is represented in {{math|SO(2''n'')}} by the preimages under the standard injection {{math|SO(2''n'') → SO(2''n'' + 1)}} of the representatives for the Weyl group of {{math|SO(2''n'' + 1)}}. Those matrices with an odd number of <math>\begin{bmatrix}
  0 & 1 \\
  1 & 0
\end{bmatrix}</math> blocks have no remaining final {{math|−1}} coordinate to make their determinants positive, and hence cannot be represented in {{math|SO(2''n'')}}.

==Topology==
{{confusing section|reason=most notations are undefined; no context for explaining why these consideration belong to the article. Moreover, the section consists esentially in a list of advanced results without providing the information that is needed for a non-specialist for verifying them (no reference, no link to articles about the methods of computation that are used, no sketch of proofs|date=November 2019}}
{{technical |section|date=November 2019}}.

===Low-dimensional topology===
The low-dimensional (real) orthogonal groups are familiar [[topological space|space]]s:
* {{math|1=O(1) = ''S''<sup>0</sup>}}, a [[2 (number)|two]]-point [[discrete topology|discrete space]]
* {{math|1=SO(1) = {1} }}
* {{math|SO(2)}} is [[circle|{{math|''S''<sup>1</sup>}}]]
* [[Rotation group SO(3)|{{math|SO(3)}}]] is [[real projective space|{{math|'''R'''P<sup>3</sup>}}]] <ref>{{harvnb|Hall|2015}} Section 1.3.4</ref>
* {{math|SO(4)}} is [[double cover (topology)|doubly covered]] by {{math|1=[[special unitary group|SU]](2) × SU(2) = [[3-sphere|''S''<sup>3</sup>]] × ''S''<sup>3</sup>}}.

===Fundamental group===
In terms of [[algebraic topology]], for {{math|''n'' > 2}} the [[fundamental group]] of {{math|SO(''n'', '''R''')}} is [[Cyclic group|cyclic of order 2]],<ref>{{harvnb|Hall|2015}} Proposition 13.10</ref> 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 connected [[double cover (topology)|2-fold cover]]).

===Homotopy groups===
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:
:<math>\operatorname{O}(0) \subset \operatorname{O}(1)\subset \operatorname{O}(2) \subset \cdots \subset O = \bigcup_{k=0}^\infty \operatorname{O}(k)</math>

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]]:
: <math>\operatorname{O}(n) \to \operatorname{O}(n + 1) \to S^n,</math>

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>(O(''n'' + 1)) = 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.

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 need only to list the lower 8 homotopy groups:

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

====Relation to KO-theory====
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:
: <math>\begin{align}
  \pi_0 (KO) &= \mathbf{Z}\\
  \pi_1 (KO) &= \mathbf{Z}/2\mathbf{Z}\\
  \pi_2 (KO) &= \mathbf{Z}/2\mathbf{Z}\\
  \pi_3 (KO) &= 0\\
  \pi_4 (KO) &= \mathbf{Z}\\
  \pi_5 (KO) &= 0\\
  \pi_6 (KO) &= 0\\
  \pi_7 (KO) &= 0
\end{align}</math>

====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.

*{{math|1=π<sub>0</sub>(''O'') = π<sub>0</sub>(O(1)) = '''Z'''/2'''Z'''}}, from [[orientation (mathematics)|orientation]]-preserving/reversing (this class survives to {{math|O(2)}} and hence stably)
*{{math|1=π<sub>1</sub>(''O'') = π<sub>1</sub>(SO(3)) = '''Z'''/2'''Z'''}}, which is [[spin group|spin]] comes from {{math|1=SO(3) = '''R'''P<sup>3</sup> = ''S''<sup>3</sup>/('''Z'''/2'''Z''')}}.
*{{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.

=====Lie groups=====
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]]).

=====Vector bundles=====
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]].

=====Loop spaces=====
Using concrete descriptions of the loop spaces in [[Bott periodicity]], one can interpret the higher homotopies of {{math|''O''}} in terms of simpler-to-analyze homotopies of lower order. 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.

====Interpretation of homotopy groups====
In a nutshell:<ref>[http://math.ucr.edu/home/baez/week105.html John Baez "This Week's Finds in Mathematical Physics" week 105]</ref>

*{{math|1=π<sub>0</sub>(''K''O) = '''Z'''}} is about [[Hamel dimension|dimension]]
*{{math|1=π<sub>1</sub>(''K''O) = '''Z'''/2'''Z'''}} is about [[orientation (mathematics)|orientation]]
*{{math|1=π<sub>2</sub>(''K''O) = '''Z'''/2'''Z'''}} is about [[spin group|spin]]
*{{math|1=π<sub>4</sub>(''K''O) = '''Z'''}} is about [[topological quantum field theory]].

Let {{math|''R''}} be any of the four [[division algebra]]s {{math|'''R'''}}, {{math|'''C'''}}, {{math|'''[[quaternions|H]]'''}}, {{math|'''[[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 {{math|''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|'''R'''P<sup>1</sup>]] = ''S''<sup>1</sup>}}, {{math|1=[[Riemann sphere|'''C'''P<sup>1</sup>]] = ''S''<sup>2</sup>}}, {{math|1='''H'''P<sup>1</sup> = ''S''<sup>4</sup>}}, {{math|1='''O'''P<sup>1</sup> = ''S''<sup>8</sup>}}, these yield vector bundles over the corresponding spheres, and

*{{math|π<sub>1</sub>(''K''O)}} is generated by {{math|[''L''<sub>'''R'''</sub>]}}
*{{math|π<sub>2</sub>(''K''O)}} is generated by {{math|[''L''<sub>'''C'''</sub>]}}
*{{math|π<sub>4</sub>(''K''O)}} is generated by {{math|[''L''<sub>'''H'''</sub>]}}
*{{math|π<sub>8</sub>(''K''O)}} is generated by {{math|[''L''<sub>'''O'''</sub>]}}

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)}}.

====Whitehead tower====
The orthogonal group anchors a [[Whitehead tower]]:

:<math>\ldots \rightarrow \operatorname{Fivebrane}(n) \rightarrow \operatorname{String}(n) \rightarrow \operatorname{Spin}(n) \rightarrow \operatorname{SO}(n) \rightarrow \operatorname{O}(n)</math>

which is obtained by successively removing (killing) homotopy groups of increasing order. This is done by constructing [[short exact sequence]]s starting with an [[Eilenberg&ndash;MacLane space]] for the homotopy group to be removed. The first few entries in the tower are the [[spin group]] and the [[string group]], and are preceded by the [[fivebrane group]].  The homotopy groups that are killed are in turn {{pi}}<sub>0</sub>(''O'') to obtain ''SO'' from ''O'', {{pi}}<sub>1</sub>(''O'') to obtain ''Spin'' from ''SO'', {{pi}}<sub>3</sub>(''O'') to obtain ''String'' from ''Spin'', and then {{pi}}<sub>7</sub>(''O'') and so on to obtain the higher order [[brane]]s.

== Of indefinite quadratic form over the reals==
{{Main|Indefinite orthogonal group}}

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

The orthogonal group of a quadratic form depends only on the inertia, and is thus generally denoted {{math|O(''p'', ''q'')}}. Moreover, as a quadratic form and its opposite have the same orthogonal group, one has {{math|1=O(''p'', ''q'') = O(''q'', ''p'')}}. 

The standard orthogonal group is {{math|1=O(''n'') = O(''n'', 0) = O(0, ''n'')}}. So, in the remainding of this section, it is supposed that neither {{mvar|p}} nor {{mvar|q}} is zero.

The subgroup of the matrices of determinant 1 in {{math|O(''p'', ''q'')}} is denoted {{math|SO(''p'', ''q'')}}. The group {{math|O(''p'', ''q'')}} has four connected components depending whether an element preserves or not the orientation of either maximal subspaces where the quadratic form is positive definite or negative definite. The component of the identity whose elements preserve the orientation of either subspaces is denoted {{math|SO{{sup|+}}(''p'', ''q'')}}.

The group {{math|O(3, 1)}} is the [[Lorentz group]] that is fundamental in [[relativity theory]]. Here the {{math|3}} corresponds to space coordinates, and {{math|1}} corresponds to the time.

==Of complex quadratic forms==
Over the field {{math|'''C'''}} of [[complex number]]s, every non-degenerate [[quadratic form]] is a sum of squares. This means that, if {{mvar|q}} is a quadratic form over a vector space {{mvar|V}} of dimension {{mvar|n}}, there are bases of {{mvar|V}} on which the matrix of {{mvar|q}} is the identity matrix, and the value of {{mvar|q}} on a vector {{math|''v'' ∈ ''V''}} is the sum of the squares of the components of {{mvar|v}}.

There is thus only one orthogonal group for each dimension over the complexes, that is usually denoted {{math|O(''n'', '''C''')}}. It can be identified with the group of ''complex orthogonal matrices'', that is the complex matrices whose product with their transpose is the identity matrix.

Similarly as in the real case, {{math|O(''n'', '''C''')}} has two connected components. The component of the identity consists of all matrices of {{math|O(''n'', '''C''')}} with 1 as their determinant, and is denoted {{math|SO(''n'', '''C''')}}.

{{math|O(''n'', '''C''')}} and {{math|SO(''n'', '''C''')}} are complex Lie groups of dimension {{math|''n''(''n'' − 1)/2}} over {{math|'''C'''}} (the dimension over {{math|'''R'''}} is twice that). For {{math|''n'' ≥ 2}} these groups are noncompact.
Just as in the real case {{math|SO(''n'', '''C''')}} is not simply connected. For {{math|''n'' &gt; 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]].

==Over finite fields==
===Characteristic different from two===
Over a field of characteristic different from two, two [[quadratic form]]s are ''equivalent'' if their matrices are [[congruent matrices|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 precicely, [[Witt's decomposition theorem]] asserts that (in characteristic different from two) every vector space equipped with a non-degenerate quadratic form {{mvar|Q}} can be decomposed as a direct sum of pairwise orthogonal subspaces
: <math>V = L_1 \oplus L_2 \oplus \cdots \oplus L_m \oplus W,</math>
where each {{mvar|L{{sub|i}}}} is a [[hyperbolic plane (quadratic forms)|hyperbolic plane]] (that is there is a basis such that the matrix of the restriction of {{mvar|Q}} to {{mvar|L{{sub|i}}}} has the form <math>\textstyle\begin{bmatrix}0&1\\1&0\end{bmatrix}</math>), and the restriction of {{mvar|Q}} to {{mvar|W}} is [[anisotropic quadratic form|anisotropic]] (that is, {{math|''Q''(''w'') ≠ 0}} for every nonzero {{mvar|w}} in {{mvar|W}}).

[[Chevalley–Warning theorem]] asserts that, over a [[finite field]] the dimension of {{mvar|W}} is at most two. 

If the dimension of {{mvar|V}} is odd, the dimension of {{mvar|W}} is thus equal to one, and its matrix is congruent either to <math>\textstyle\begin{bmatrix}1\end{bmatrix}</math> or to <math>\textstyle\begin{bmatrix}\varphi\end{bmatrix},</math> where {{mvar|{{phi}}}} is a non-square scalar. It results that there is only one orthogonal group that is denoted {{math|O(2''n'' + 1, ''q'')}}, where {{mvar|q}} is the number of elements of the finite field (a power of an odd prime).<ref name=Wil6975>{{cite book | last=Wilson | first=Robert A. | title=The finite simple groups | zbl=1203.20012 | series=Graduate Texts in Mathematics | volume=251 | location=London | publisher=Springer | isbn=978-1-84800-987-5 | year=2009 | pages=69–75 }}</ref>

If the dimension of {{mvar|W}} is two and {{math|–1}} is not a square in the ground field (that is, if its number of elements {{mvar|q}} is congruent to 3 modulo 4), the matrix of the restriction of {{mvar|Q}} to {{mvar|W}} is congruent to either {{mvar|I}} or {{math|–''I''}}, where {{mvar|I}} is the 2×2 identity matrix. If the dimension of {{mvar|W}} is two and {{math|–1}} is a square in the ground field (that is, if {{mvar|q}} is congruent to 1, modulo 4) the matrix of the restriction of {{mvar|Q}} to {{mvar|W}} is congruent to <math>\textstyle\begin{bmatrix}1&0\\0&\phi\end{bmatrix},</math> {{mvar|{{phi}}}} is any non-square scalar.

This implies that if the dimension of {{mvar|V}} is even, there are only two orthogonal groups, depending whether the dimension of {{mvar|W}} zero or two. They are denoted respectively {{math|O<sup>+</sup>(2''n'', ''q'')}} and {{math|O<sup>−</sup>(2''n'', ''q'')}}.<ref name=Wil6975 />

The orthogonal group {{math|O<sup>''ϵ''</sup>(2, ''q'')}} is a [[dihedral group]] of order {{math|2(''q''  − ''ϵ'')}}, where {{math|1=''ϵ'' = ±}}.
{{cot|Proof}}
For studying the orthogonal group of {{math|O<sup>''ϵ''</sup>(2, ''q'')}}, one can suppose that the matrix of the quadratic form is <math>Q=\begin{bmatrix}1&0\\0&-\omega\end{bmatrix},</math> because, given a quadratic form, there is a basis where its matrix is diagonalizable. A matrix <math>A=\begin{bmatrix}a&b\\c&d\end{bmatrix}</math> belongs to the orthogonal group if <math>AQA^T=Q,</math> that is, {{math|1= ''a''{{sup|2}} – ''ωb''{{sup|2}} = 1}}, {{math|1=''ac'' – ''ωbd'' = 0}}, and {{math|1= ''c''{{sup|2}} – ''ωd''{{sup|2}} = –ω}}. As {{mvar|a}} and {{mvar|b}} cannot be both zero (because of the first equation), the second equation implies the existence of {{mvar|ϵ}} in {{math|'''F'''{{sub|''q''}}}}, such that {{math|1=''c'' = ''ϵωb''}} and {{math|1=''d'' = ''ϵa''}}. Reporting these values in the third equation, and using the first equation, one gets that {{math|1=''ϵ''{{sup|2}} = 1 }}, and thus the orthogonal group consists of the matrices 
:<math>\begin{bmatrix}a&b\\\epsilon\omega b&\epsilon a\end{bmatrix},</math>
where {{math|1= ''a''{{sup|2}} – ''ωb''{{sup|2}} = 1}} and {{math|1=ϵ = ±1}}. Moreover, the determinant of the matrix is {{mvar|ϵ}}. 

For further studying the orthogonal group, it is convenient to introduce a square root {{mvar|α}} of {{mvar|ω}}. This square root belongs to {{math|'''F'''{{sub|''q''}}}} if the orthogonal group is {{math|O<sup>+</sup>(2, ''q'')}}, and to {{math|'''F'''{{sub|''q''{{sup|2}}}}}} otherwise. Setting {{math|1=''x'' = ''a'' + ''αb''}}, and {{math|1=''y'' = ''a'' – ''αb''}}, one has
:<math>xy=1,\qquad a=\frac{x+y}2\qquad b=\frac{x-y}{2\alpha}.</math>

If <math>A_1=\begin{bmatrix}a_1&b_1\\\omega b_1&a_1\end{bmatrix}</math> and <math>A_2=\begin{bmatrix}a_2&b_2\\\omega b_2& a_2\end{bmatrix}</math> are two matrices of determinant one in the orthogonal group then 
:<math>A_1A_2=\begin{bmatrix}a_1a_2+\omega b_1b_2 & a_1b_2+ b_1 a_2\\
\omega b_1a_2 +\omega a_1b_2 & \omega b_1b_2+ a_1a_1\end{bmatrix}. </math>
This is an orthogonal matrix <math>\begin{bmatrix}a&b\\\omega b&a\end{bmatrix},</math>
with  {{math|1=''a'' = ''a''{{sub|1}}''a''{{sub|2}} + ''ωb''{{sub|1}}''b''{{sub|2}}}}, and {{math|1=''b'' = ''a''{{sub|1}}''b''{{sub|2}} + ''b''{{sub|1}}''a''{{sub|2}}}}. Thus 
:<math>a+\alpha b = (a_1+\alpha b_1)(a_2+\alpha b_2).</math>

It follows that the map <math>(a,b)\mapsto a+\alpha b</math> is a homomorphism of the group of orthogonal matrices  of determinant one into the multiplicative group of {{math|'''F'''{{sub|''q''{{sup|2}}}}}}. 

In the case of {{math|O<sup>+</sup>(2''n'', ''q'')}}, the image is the multiplicative group of {{math|'''F'''{{sub|''q''}}}}, which is a cyclic group of order {{mvar|q}}. 

In the case of {{math|O<sup>–</sup>(2''n'', ''q'')}}, the above {{mvar|x}} and {{mvar|y}} are [[Conjugate element (field theory)|conjugate]], and are therefore the image of each other by the [[Frobenius automorphism]]. This meand that <math>y=x^{-1}=x^q,</math> and thus <math>x^{q+1}=1.</math> For every such {{mvar|x}} one can reconstruct a corresponding orthogonal matrix. It follows that the map <math>(a,b)\mapsto a+\alpha b</math> is a group isomorphism from the orthogonal matrices of determinant 1 to the group of the {{math|(''q'' + 1)}}-[[roots of unity]]. This group is a cyclic group of order {{math|''q'' + 1}} which consists of the powers of <math>g^{q-1},</math> where {{mvar|g}} is a [[primitive element (finite field)|primitive element]] of {{math|'''F'''{{sub|''q''{{sup|2}}}}}}, 

For finishing the proof, it suffices to verify that the group all orthogonal natrices is not abelian, and is the semidirect product of the group {{math|{{mset|1, –1}}}} and the group of orthogonal matrices of determinant one.

The comparison of this proof with the real case may be illuminating.

Here two group isomorphisms are involved:
:<math>\begin{align}
\mathbb Z/(q+1)\mathbb Z &\to T\\
k&\mapsto g^{(q-1)k},
\end{align}</math>
where {{mvar|g}} is a primitive element of {{math|'''F'''{{sub|''q''{{sup|2}}}}}} and {{mvar|T}} is the multiplicative group of the element of norm one in {{math|'''F'''{{sub|''q''{{sup|2}}}}}} ;
:<math>\begin{align}
\mathbb T &\to \operatorname{SO}^+ (2, \mathbf F_q) \\
x&\mapsto \begin{bmatrix}a&b\\\omega b&a\end{bmatrix},
\end{align}</math>
with <math>a = \frac {x+x^{-1}}2</math> and <math>b = \frac {x- x^{-1}}{2\alpha}.</math>

In the real case, the corresponding isomorphisms are:
:<math>\begin{align}
\mathbb R/2\pi\mathbb R &\to C\\
\theta&\mapsto e^{i\theta},
\end{align}</math>
where {{mvar|C}} is the circle of the complex numbers of norm one;
:<math>\begin{align}
\mathbb C &\to \operatorname{SO}(2, \mathbb R) \\
x&\mapsto \begin{bmatrix}\cos\theta&\sin\theta\\-\sin\theta&\cos\theta\end{bmatrix},
\end{align}</math>
with <math>\cos\theta = \frac {e^{i\theta}+e^{-i\theta}}2</math> and <math>\sin\theta = \frac {e^{i\theta}-e^{-i\theta}}{2i}.</math>

{{cob}}

When the characteristic is not two, the order of the ortogonal groups are{{cn|date=November 2019}}{{clarify|reason=If the order is the same for O+ and O– this must be explicitly staded. Otherwise, as suggested by the case n=1, four cases seem to be needed|date=November 2019}}
: <math>\left|\operatorname{O}(2n + 1, q)\right| = 2q^n\prod_{i=0}^{n-1}\left(q^{2n} - q^{2i}\right),</math>
: <math>\left|\operatorname{O}(2n, q)\right| = 2\left(q^n - 1\right)\prod_{i=1}^{n-1}\left(q^{2n} - q^{2i}\right),\qquad\qquad\quad\;</math> if {{math|−1}} is a square ({{math|1=''q'' = 4''k'' + 1}}),
:<math>\left|\operatorname{O}(2n, q)\right| = 2\left(q^n + (-1)^{n+1}\right)\prod_{i=1}^{n-1}\left(q^{2n} - q^{2i}\right),\qquad</math> if {{math|−1}} is not a square ({{math|1=''q'' = 4''k'' + 3}}).

===The Dickson invariant===
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 {{math|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>

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.
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 (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 {{math|''F''}} is not 2, the Dickson Invariant is {{math|0}} whenever the determinant is {{math|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 {{math|1}}. Each element in {{math|O(''n'', ''F'' )}} has determinant {{math|±1}}. Thus in characteristic 2, the determinant is always {{math|1}}.

The Dickson invariant can also be defined for [[Clifford group]]s and [[Pin group]]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.<ref>{{harv|Grove|2002|loc=Theorem 6.6 and 14.16}}</ref> 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'''}}.
*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''}}.
*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.
*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<!-- This section is linked from [[Clifford algebra]] -->==
The '''spinor norm''' is a homomorphism from an orthogonal group over a field {{math|''F''}} to the [[quotient group]] {{math|''F''<sup>×</sup>/(''F''<sup>×</sup>)<sup>2</sup>}} (the [[multiplicative group]] of the field {{math|''F''}} [[up to]] multiplication by [[square (algebra)|square]] elements), that takes reflection in a vector of norm {{math|''n''}} to the image of {{math|''n''}} in {{math|''F''<sup>×</sup>/(''F''<sup>×</sup>)<sup>2</sup>}}.<ref name=C178>{{harvnb|Cassels|1978|p=178}}</ref>

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

==Galois cohomology and orthogonal groups==
In the theory of [[Galois cohomology]] of [[algebraic 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]].

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.

:<math> 1 \rightarrow \mu_2 \rightarrow \mathrm{Pin}_V \rightarrow \mathrm{O_V} \rightarrow 1 </math>

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 {{math|''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.

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.

==Lie algebra==
{{anchor|orthogonal Lie algebra}}{{anchor|special orthogonal Lie algebra}}
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 {{math|''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''}}.

Since the group {{math|SO(''n'')}} 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 {{math|SO(''n'')}} are just linear representations of the universal cover, the [[spin group]] Spin(''n'').) The latter are the so-called [[spin representation]], which are important in physics.

More intrinsically, given a vector space with an inner product{{clarify|reason=This is presumably intended to be a nondegenerate bilinear form, not an [[inner product]]|date=March 2017}}, 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'''<sup>∗</sup>}} is the [[covector]] dual to the vector {{math|'''v'''}};{{clarify|reason=Given a nondegenerate bilinear form, there is a natural map from the vectors onto covectors (see Bilinear_form#Maps_to_the dual_space); the term "covector dual to the vector" is inappropriate though and should be rephased.|date=March 2017}} in coordinates these are exactly the elementary skew-symmetric{{clarify|reason=This should presumably be "alternating", to cover the case of fields of characteristic 2.|date=March 2017}} matrices.

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>

==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 {{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]].

===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:

:<math>\mathrm{O}(n) \supset \mathrm{O}(n - 1)</math> – preserve an axis
:<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.
:<math>\mathrm{O}(2n) \supset \mathrm{USp}(n)</math>
:<math>\mathrm{O}(7) \supset \mathrm{G}_2</math>

===Lie supergroups===
The orthogonal group {{math|O(''n'')}} is also an important subgroup of various Lie groups:

:<math>\begin{align}
     \mathrm{U}(n) \supset \mathrm{SU}(n) &\supset \mathrm{O}(n) \\
                         \mathrm{USp}(2n) &\supset \mathrm{O}(n) \\
                             \mathrm{G}_2 &\supset \mathrm{O}(3) \\
                             \mathrm{F}_4 &\supset \mathrm{O}(9) \\
                             \mathrm{E}_6 &\supset \mathrm{O}(10) \\
                             \mathrm{E}_7 &\supset \mathrm{O}(12) \\
                             \mathrm{E}_8 &\supset \mathrm{O}(16)
\end{align}</math>

====Conformal group====
{{Main|Conformal group}}

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 [[Homothetic transformation|dilations]]. If {{math|''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'''<sup>∗</sup>}}, where {{math|1='''R'''<sup>∗</sup> = '''R'''∖{0}}} is the real [[multiplicative group]], while if {{math|''n''}} is even, these subgroups intersect in {{math|±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>}}.

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>}}.

===Discrete subgroups===
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]]s 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.

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]].

Other finite subgroups include:
* [[Permutation matrices]] (the [[Coxeter group]] {{math|A<sub>''n''</sub>}})
* [[Signed permutation matrices]] (the [[Coxeter group]] {{math|B<sub>''n''</sub>}}); also equals the intersection of the orthogonal group with the [[integer matrix|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 {{math|±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>

===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 group]]s, {{math|Pin<sub>+</sub>(''n'') → O(''n'')}} and {{math|Pin<sub>−</sub>(''n'') → O(''n'')}},
* The quotient [[projective orthogonal group]], {{math|O(''n'') → PO(''n'')}}.
These are all 2-to-1 covers.

For the special orthogonal group, the corresponding groups are:
* [[Spin group]], {{math|Spin(''n'') → SO(''n'')}},
* [[Projective special orthogonal group]], {{math|SO(''n'') → PSO(''n'')}}.
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>

In dimension 3 and above these are the covers and quotients, while dimension 2 and below are somewhat degenerate; see specific articles for details.

==Principal homogeneous space: Stiefel manifold==
{{Main|Stiefel manifold}}
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 basis|orthonormal bases]] (orthonormal [[k-frame|{{math|''n''}}-frames]]).

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

The other Stiefel manifolds {{math|''V<sub>k</sub>''('''R'''<sup>''n''</sup>)}} for {{math|''k'' < ''n''}} of ''incomplete'' orthonormal bases (orthonormal {{math|''k''}}-frames) are still homogeneous spaces for the orthogonal group, but not ''principal'' homogeneous spaces: any {{math|''k''}}-frame can be taken to any other {{math|''k''}}-frame by an orthogonal map, but this map is not uniquely determined.

==See also==

===Specific transforms===
*[[Coordinate rotations and reflections]]
*[[Reflection through the origin]]

===Specific groups===
*rotation group, [[Rotation group SO(3)|SO(3, '''R''')]]
*[[SO(8)]]

===Related groups===
*[[indefinite orthogonal group]]
*[[unitary group]]
*[[symplectic group]]

===Lists of groups===
*[[list of finite simple groups]]
*[[list of simple Lie groups]]

==Notes==
{{Reflist| group = note }}

==Citations==
{{Reflist}}

==References==
{{refbegin}}
* {{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 }}
* {{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}}
* {{Citation| last=Hall|first=Brian C.|title=Lie Groups, Lie Algebras, and Representations: An Elementary Introduction|edition=2nd|series=Graduate Texts in Mathematics|volume=222|publisher=Springer|year=2015|isbn=978-3319134666}}
* {{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 }}
{{refend}}

==External links==
*{{springer|title=Orthogonal group|id=p/o070300}}
*[http://math.ucr.edu/home/baez/week105.html John Baez "This Week's Finds in Mathematical Physics" week 105]
*[http://math.ucr.edu/home/baez/octonions/node10.html John Baez on Octonions]
* {{it}} [http://ansi.altervista.org n-dimensional Special Orthogonal Group parametrization]

[[Category:Lie groups]]
[[Category:Quadratic forms]]
[[Category:Euclidean symmetries]]
[[Category:Linear algebraic groups]]