Editing Orthogonal group (section)

=== 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 complicated structure (in particular, it is no longer 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.