Editing 3D rotation group (section)

==Group structure==
The rotation group is a [[group (mathematics)|group]] under [[function composition]] (or equivalently the [[matrix product|product of linear transformations]]). It is a [[subgroup]] of the [[general linear group]] consisting of all [[invertible matrix|invertible]] linear transformations of the [[real coordinate space|real 3-space]] <math>\R^3</math>.<ref>''n''&nbsp;×&nbsp;''n'' real matrices are identical to linear transformations of <math>\R^n</math> expressed in its [[standard basis]].</ref>

Furthermore, the rotation group is [[nonabelian group|nonabelian]]. That is, the order in which rotations are composed makes a difference. For example, a quarter turn around the positive ''x''-axis followed by a quarter turn around the positive ''y''-axis is a different rotation than the one obtained by first rotating around ''y'' and then ''x''.

The orthogonal group, consisting of all proper and improper rotations, is generated by reflections. Every proper rotation is the composition of two reflections, a special case of the [[Cartan–Dieudonné theorem]].

===Complete classification of finite subgroups===
The finite subgroups of <math>\mathrm{SO}(3)</math> are completely [[classification theorem|classified]].<ref name="coxeter">{{cite book |last1=Coxeter |first1=H. S. M. |title=Regular polytopes |date=1973 |location=New York |isbn=0-486-61480-8 |page=53 |edition=Third}}</ref>

Every finite subgroup is isomorphic to either an element of one of two [[countably infinite]] families of planar isometries: the [[cyclic group]]s <math>C_n</math> or the [[dihedral group]]s <math>D_{2n}</math>, or to one of three other groups: the [[tetrahedral group]] <math>\cong A_4</math>, the [[octahedral group]] <math>\cong S_4</math>, or the [[icosahedral group]] <math>\cong A_5</math>.