Editing Orthogonal group (section)

=== Special orthogonal group ===
By choosing an [[orthonormal basis]] of a 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 the determinant of {{mvar|Q}} is either {{math|1}} or {{math|−1}}. The orthogonal matrices with determinant {{math|1}} form a subgroup called the ''special orthogonal group'', denoted {{math|SO(''n'')}}, consisting 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 a group homomorphism whose image is the multiplicative group {{math|{{mset|−1, +1}}}}. This implies that the orthogonal group is an internal [[semidirect product]] of {{math|SO(''n'')}} and any subgroup  formed with the identity and a [[reflection (geometry)|reflection]].

The group with two elements {{math|{{mset|±''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|{{mset|±''I''}}}}.

The group {{math|SO(2)}} is [[abelian group|abelian]] (whereas {{math|SO(''n'')}} is not abelian when {{math|''n'' > 2}}). Its finite subgroups are the [[cyclic group]] {{math|''C''<sub>''k''</sub>}} of [[rotational symmetry|{{math|''k''}}-fold rotations]], for every positive integer {{mvar|k}}. All these groups are  normal subgroups of {{math|O(2)}} and {{math|SO(2)}}.