Editing Orthogonal group (section)

{{Short description|Type of group in mathematics}}

{{Group theory sidebar |Topological}}
In [[mathematics]], the '''orthogonal group''' in dimension {{math|''n''}}, 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. The orthogonal group is sometimes called the '''general orthogonal group''', by analogy with the [[general linear group]]. 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 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 [[normal subgroup]], called the '''special orthogonal group''', and denoted {{math|SO(''n'')}}. It consists of all orthogonal matrices of [[determinant]] 1. This group is also called the '''rotation group''', generalizing the fact that 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)]]}}. The other component consists of all orthogonal matrices of determinant {{math|−1}}.  This component does not form a group, as the product of any two of its elements is of determinant 1, and therefore not an element of the component.

By extension, for any field {{math|''F''}}, an {{math|''n'' × ''n''}} matrix with entries in {{math|''F''}} such that its inverse equals its transpose is called an ''orthogonal matrix over'' {{math|''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 display="block">\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 special 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 group]]s, since the condition of preserving a form can be expressed as an equality of matrices.