# Orthogonal group
An orthogonal group is the group of all linear transformations of a vector space that preserve a fixed non-singular quadratic form, that is, linear maps \( \phi \) with \( Q(\phi(v)) = Q(v) \) for every vector \( v \). It is one of the classical groups, and its elements are called orthogonal transformations, or automorphisms of the form \( Q \).[1](https://encyclopediaofmath.org/wiki/Orthogonal_group) In the most familiar case, the group \( O(n) \) is the group of \( n \times n \) real orthogonal matrices, the distance-preserving transformations of \( n \)-dimensional [Euclidean space](https://www.edgepedia.org/euclidean-space) that fix a chosen point.[2](https://mathworld.wolfram.com/OrthogonalGroup.html)
| Key fact | Detail |
|---|---|
| Definition | Linear transformations \( \phi \) of an \( n \)-dimensional space \( V \) over a field \( k \) preserving a non-singular quadratic form \( Q \), denoted \( O_n(k,f) \)[1](https://encyclopediaofmath.org/wiki/Orthogonal_group) |
| Matrix form | With \( B \) the matrix of the bilinear form, the group is all matrices \( A \) over \( k \) with \( A^T B A = B \)[1](https://encyclopediaofmath.org/wiki/Orthogonal_group) |
| Determinant | Every element has determinant \( 1 \) or \( -1 \); the determinant-1 rotations form a normal subgroup \( O_n^+ \) of index 2[1](https://encyclopediaofmath.org/wiki/Orthogonal_group) |
| Dimension | \( O(n) \) is a compact Lie group of dimension \( n(n-1)/2 \), with Lie algebra the skew-symmetric real \( n \times n \) matrices[2](https://mathworld.wolfram.com/OrthogonalGroup.html) |
| Algebraic group type | \( SO_n \) is almost-simple of type \( B_s \) when \( n = 2s+1 \) and type \( D_s \) when \( n = 2s \) with \( s \ge 3 \); its universal cover is a spinor group[1](https://encyclopediaofmath.org/wiki/Orthogonal_group) |
| Indefinite forms | For signature \( (p,q) \) the group is \( O(p,q) \); the Lorentz group is \( O(3,1) \)[2](https://mathworld.wolfram.com/OrthogonalGroup.html) |
## Definition and matrix description
Let \( V \) be an \( n \)-dimensional vector space over a field \( k \) of characteristic not 2, and let \( Q \) be a non-singular quadratic form on \( V \). The associated symmetric bilinear form is
\[ f(u,v) = \tfrac{1}{2}\left( Q(u+v) - Q(u) - Q(v) \right), \]
and the orthogonal group \( O_n(k,f) \) consists of the linear transformations of \( V \) that preserve \( f \).[1](https://encyclopediaofmath.org/wiki/Orthogonal_group) If \( B \) is the matrix of \( f \) in some basis, the group identifies with the set of \( n \times n \) matrices \( A \) over \( k \) satisfying
\[ A^T B A = B, \]
where \( {}^T \) denotes transposition.[1](https://encyclopediaofmath.org/wiki/Orthogonal_group) For the standard real case this reduces to \( A A^T = I \), the defining equation of the orthogonal matrices.[2](https://mathworld.wolfram.com/OrthogonalGroup.html)
Rescaling the form does not change the group: over a field, replacing the form by a scalar multiple \( \alpha (x \cdot y) \) leaves \( O(V) \) unchanged.[3](https://bmonson.ext.unb.ca/fields/orth.pdf)
## Rotations, reflections, and the determinant
The determinant of any element of \( O_n \) equals \( 1 \) or \( -1 \). Elements with determinant 1 are called rotations and form a normal subgroup \( O_n^+(k,f) \) of index 2, the rotation group; elements outside it are called inversions. Every rotation is the product of an even number of reflections, and every inversion the product of an odd number of reflections from \( O_n \).[1](https://encyclopediaofmath.org/wiki/Orthogonal_group)
The Cartan-Dieudonné theorem, a standard result in the theory of quadratic forms over fields of characteristic not 2, states that orthogonal transformations are products of reflections.[4](https://www.mun.ca/aac/mini-courses/mini-courses-2016---2025/erhard-neher/) Over finite fields of characteristic 2 the statement needs qualification: almost all quadratic spaces have the property that orthogonal transformations are products of reflections, with rotations being even products, and there is a specific exceptional space \( N \oplus N \) over \( \mathbb{F}_2 \), with quadratic form \( x^2 + xy + y^2 \), where the isometry exchanging the two copies is not a product of two reflections.[5](https://www-fourier.univ-grenoble-alpes.fr/~peters/Books/QuadraticForms/QuadForms.pdf)
## Lie group structure
For the positive definite real form, \( O(n) \) is the group of \( n \times n \) orthogonal matrices, closed under multiplication and inversion. Writing a matrix as \( n^2 \) coordinate functions, the \( n^2 \) equations \( A A^T = I \) are redundant, with only \( n(n+1)/2 \) independent, leaving \( n(n-1)/2 \) free variables; the group is a smooth \( n(n-1)/2 \)-dimensional submanifold and hence a [Lie group](https://www.edgepedia.org/lie-group).[2](https://mathworld.wolfram.com/OrthogonalGroup.html) As a topological space it is compact.[6](https://ncatlab.org/nlab/show/orthogonal%2Bgroup)
The [Lie algebra](https://www.edgepedia.org/lie-algebra) \( \mathfrak{o}(n) \) consists of the real skew-symmetric \( n \times n \) matrices.[2](https://mathworld.wolfram.com/OrthogonalGroup.html) Every such matrix can be reduced by an orthogonal change of basis to two-dimensional rotational blocks, together with a zero block when the dimension is odd.[7](https://watanabe.you/wiki/Orthogonal%20Group) The determinant splits \( O(n) \) into two components, the identity component being \( SO(n) \).[2](https://mathworld.wolfram.com/OrthogonalGroup.html) For \( O(2) \), the group consists of the plane rotations \( \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix} \) and their reflected counterparts, preserving the quadratic form \( x^2 + y^2 \) and hence the circles \( x^2 + y^2 = r^2 \); as a manifold, \( O(2) \) is two disjoint copies of the circle.[2](https://mathworld.wolfram.com/OrthogonalGroup.html)
For an indefinite form of signature \( (p,q) \), the group \( O(p,q) \) preserves \( x_1^2 + \cdots + x_p^2 - x_{p+1}^2 - \cdots - x_{p+q}^2 \). Its Lie algebra satisfies \( T = -\eta T^T \eta \) for the signature matrix \( \eta \), and \( \dim(\mathfrak{so}(p,q)) = \tfrac{1}{2}n(n-1) \) with \( n = p+q \).[8](https://www-thphys.physics.ox.ac.uk/people/AndreLukas/GroupsandRepresentations/groupsrepslecturenotes.pdf) When \( q = 0 \), \( O(p,q) \) is written \( O(n) \), the real orthogonal group.[1](https://encyclopediaofmath.org/wiki/Orthogonal_group)
## Algebraic group structure
The defining equations \( A A^T = I \) are polynomials of degree two in the entries, so \( O(n) \) is a linear algebraic group.[2](https://mathworld.wolfram.com/OrthogonalGroup.html) More generally, over a field \( k \) with algebraic closure \( \bar{k} \), the form \( f \) extends to \( V \otimes_k \bar{k} \), and \( O_n(\bar{k},f) \) is a linear algebraic group defined over \( k \) with \( O_n(k,f) \) as its group of \( k \)-points. The [special orthogonal group](https://www.edgepedia.org/special-orthogonal-group) \( SO_n(\bar{k}) \) is the connected component of the identity. It is an almost-simple algebraic group, meaning it contains no infinite algebraic normal subgroups, of type \( B_s \) when \( n = 2s+1 \) with \( s \ge 1 \), and of type \( D_s \) when \( n = 2s \) with \( s \ge 3 \). The universal covering group of \( SO_n \) is a spinor group.[1](https://encyclopediaofmath.org/wiki/Orthogonal_group)
## Orthogonal groups over finite fields
Over a finite field \( \mathbb{F}_q \) of odd characteristic, the discriminant classifies quadratic forms of a given dimension up to isometry, and there is one orthogonal group in odd dimension and two in even dimension.[9](https://surim.stanford.edu/sites/g/files/sbiybj26191/files/media/file/surim_2025_paper_rawley_miguel_0.pdf) In even dimension the two classes are distinguished by a parameter \( \varepsilon = +1 \) or \( -1 \): the plus type has an isotropic basis, making the plane a hyperbolic plane, while the minus type does not.[3](https://bmonson.ext.unb.ca/fields/orth.pdf) In characteristic 2 the discriminant is not a useful invariant for even-dimensional forms, and the classifying invariant is instead the Arf invariant, which over finite fields of characteristic 2 classifies quadratic spaces of even rank.[9](https://surim.stanford.edu/sites/g/files/sbiybj26191/files/media/file/surim_2025_paper_rawley_miguel_0.pdf)
Characteristic 2 also changes the algebra: the bilinear form \( f(x,y) = Q(x+y) + Q(x) + Q(y) \) is alternating rather than symmetric, so the orthogonal group sits inside a symplectic group.[10](https://mathoverflow.net/questions/74090/the-maximal-order-of-an-element-in-orthogonal-groups-over-finite-fields-of-chara) Small cases can be computed directly: \( O_2^{\pm}(q) \cong D_{2(q \mp 1)} \), the dihedral group of order \( 2(q \mp 1) \), so over \( \mathbb{F}_2 \) the group \( O_2^+(2) \) is cyclic of order 2 and \( O_2^-(2) \) is dihedral of order 6.[11](https://mathoverflow.net/questions/178466/orthogonal-group-in-characteristic-2) Over larger fields the groups stay finite: \( O(2, \mathbb{F}_{23}) \) has 48 elements.[2](https://mathworld.wolfram.com/OrthogonalGroup.html)
## Role in physics
Orthogonal groups appear in physics both as global and as gauge symmetries. The group \( SO(10) \) is a prominent unification group for the standard model of particle physics, and \( (S)O(n-1,1) \) is the [Lorentz group](https://www.edgepedia.org/lorentz-group) in \( n \) dimensions, which underlies \( n \)-dimensional relativistic theories in the same way \( SO(3,1) \) underlies four-dimensional covariant physics; the groups \( SO(p,q) \) also appear as string duality symmetries.[8](https://www-thphys.physics.ox.ac.uk/people/AndreLukas/GroupsandRepresentations/groupsrepslecturenotes.pdf) University physics curricula accordingly give the unitary and orthogonal groups, and in particular the rotation and Lorentz groups, a central place in Lie group theory courses.[12](https://www.unibo.it/en/study/course-units-transferable-skills-moocs/course-unit-catalogue/course-unit/2026/535146)
## References
1. [Orthogonal group, Encyclopedia of Mathematics (entry by V.L. Popov)](https://encyclopediaofmath.org/wiki/Orthogonal_group)
2. [Orthogonal Group, Wolfram MathWorld](https://mathworld.wolfram.com/OrthogonalGroup.html)
3. [B. Monson, Orthogonal groups over finite fields](https://bmonson.ext.unb.ca/fields/orth.pdf)
4. [E. Neher, Quadratic Forms over Fields, mini-course, Atlantic Algebra Centre, Memorial University of Newfoundland](https://www.mun.ca/aac/mini-courses/mini-courses-2016---2025/erhard-neher/)
5. [Quadratic Forms, Chapter 16: The Structure of Orthogonal Groups I (Peters, Université Grenoble Alpes)](https://www-fourier.univ-grenoble-alpes.fr/~peters/Books/QuadraticForms/QuadForms.pdf)
6. [Orthogonal group, nLab](https://ncatlab.org/nlab/show/orthogonal%2Bgroup)
7. [Orthogonal Group, Watanabe reference wiki](https://watanabe.you/wiki/Orthogonal%20Group)
8. [A. Lukas, Groups and Representations, lecture notes, Oxford Theoretical Physics](https://www-thphys.physics.ox.ac.uk/people/AndreLukas/GroupsandRepresentations/groupsrepslecturenotes.pdf)
9. [R. Harrison and M. Machado, On Finite Subgroups of Orthogonal Groups, SURIM 2025, Stanford](https://surim.stanford.edu/sites/g/files/sbiybj26191/files/media/file/surim_2025_paper_rawley_miguel_0.pdf)
10. [The maximal order of an element in orthogonal groups over finite fields of characteristic 2, MathOverflow](https://mathoverflow.net/questions/74090/the-maximal-order-of-an-element-in-orthogonal-groups-over-finite-fields-of-chara)
11. [Orthogonal group in characteristic 2, MathOverflow](https://mathoverflow.net/questions/178466/orthogonal-group-in-characteristic-2)
12. [Group Theory for Physics, University of Bologna course catalogue 2026/2027](https://www.unibo.it/en/study/course-units-transferable-skills-moocs/course-unit-catalogue/course-unit/2026/535146)
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