Editing Improper rotation (section)

== Three dimensions==
{{See|Point groups in three dimensions}}
In 3 dimensions, improper rotation is equivalently defined as a combination of rotation about an axis and [[inversion in a point]] on the axis.<ref name="Morawiec"/> For this reason it is also called a '''rotoinversion''' or '''rotary inversion'''. The two definitions are equivalent because [[angle of rotation|rotation by an angle]] θ followed by reflection is the same transformation as rotation by θ&nbsp;+&nbsp;180° followed by inversion (taking the point of inversion to be in the plane of reflection). In both definitions, the operations commute.

A three-dimensional symmetry that has only one [[Fixed points of isometry groups in Euclidean space|fixed point]] is necessarily an improper rotation.<ref name="sss"/> 

An improper rotation of an object thus produces a rotation of its [[mirror image]]. The axis is called the '''rotation-reflection axis'''.<ref name="Bishop">{{citation|title=Group Theory and Chemistry|first=David M.|last=Bishop|publisher=Courier Dover Publications|year=1993|isbn=978-0-486-67355-4|page=13|url=https://books.google.com/books?id=l4zv4dukBT0C&pg=PA13}}.</ref> This is called an '''''n''-fold improper rotation''' if the angle of rotation, before or after reflexion, is 360°/''n'' (where ''n'' must be even).<ref name="Bishop"/> There are several different systems for naming individual improper rotations:
* In the [[Schoenflies notation]] the symbol '''''S<sub>n</sub>''''' (German, ''{{lang|de|Spiegel}}'', for ''[[mirror]]''), where ''n'' must be even, denotes the symmetry group generated by an ''n''-fold improper rotation. For example, the symmetry operation ''S''<sub>6</sub> is the combination of a rotation of (360°/6)=60° and a mirror plane reflection. (This should not be confused with the same notation for [[symmetric group]]s).<ref name="Bishop"/> 
* In [[Hermann–Mauguin notation]] the symbol {{overline|''n''}} is used for an '''''n''-fold rotoinversion'''; i.e., rotation by an angle of rotation of 360°/''n'' with inversion. If ''n'' is even it must be divisible by 4. (Note that {{overline|2}} would be simply a reflection, and is normally denoted "m", for "mirror".) When ''n'' is odd this corresponds to a 2''n''-fold improper rotation (or rotary reflexion).
* The [[Coxeter notation]] for ''S''<sub>2''n''</sub> is [2''n''<sup>+</sup>,2<sup>+</sup>] and {{CDD|node_h2|2x|n|node_h4|2x|node_h2}}, as an index 4 subgroup of [2''n'',2], {{CDD|node|2x|n|node|2|node}}, generated as the product of 3 reflections.
* The [[Orbifold notation]] is ''n''×, order 2''n''.[[File:Rotoreflection_subgroup_tree.png|thumb|Subgroups for ''S''<sub>2</sub> to ''S''<sub>20</sub>.<BR>''C''<sub>1</sub> is the [[Trivial group|identity group]].<BR>''S''<sub>2</sub> is the [[central inversion]].<BR>''C''<sub>n</sub> are [[cyclic group]]s.]]
=== Subgroups===
* The [[Coxeter notation#Subgroups|direct subgroup]] of ''S''<sub>2''n''</sub> is ''C''<sub>''n''</sub>, order ''n'', [[index of a subgroup|index]] 2, being the rotoreflection generator applied twice.
* For odd ''n'', ''S''<sub>2''n''</sub> contains an [[Inversion in a point|inversion]], denoted ''C''<sub>i</sub> or ''S''<sub>2</sub>. ''S''<sub>2''n''</sub> is the [[direct product]]: ''S''<sub>2''n''</sub> = ''C''<sub>''n''</sub>&nbsp;×&nbsp;''S''<sub>2</sub>, if ''n'' is odd.
* For any ''n'', if odd ''p'' is a divisor of ''n'', then ''S''<sub>2''n''/''p''</sub> is a subgroup of ''S''<sub>2''n''</sub>, index ''p''. For example ''S''<sub>4</sub> is a subgroup of ''S''<sub>12</sub>, index 3.