Editing Dihedral group (section)

==Small dihedral groups==
[[File:Regular hexagon symmetries.svg|thumb|350px|Example subgroups from a hexagonal dihedral symmetry]]
{{math|D{{sub|1}}}} is [[isomorphic]] to {{math|Z{{sub|2}}}}, the [[cyclic group]] of order 2.

{{math|D{{sub|2}}}} is [[isomorphic]] to {{math|K{{sub|4}}}}, the [[Klein four-group]].

{{math|D{{sub|1}}}} and {{math|D{{sub|2}}}} are exceptional in that:
* {{math|D{{sub|1}}}} and {{math|D{{sub|2}}}} are the only [[abelian group|abelian]] dihedral groups. Otherwise, {{math|D{{sub|''n''}}}} is non-abelian.
* {{math|D{{sub|''n''}}}} is a [[subgroup]] of the [[symmetric group]] {{math|S{{sub|''n''}}}} for {{math|''n'' ≥ 3}}. Since {{math|2''n'' > ''n''!}} for  {{math|''n'' {{=}} 1}} or {{math|''n'' {{=}} 2}}, for these values, {{math|D{{sub|''n''}}}} is too large to be a subgroup.
* The inner automorphism group of {{math|D{{sub|2}}}} is trivial, whereas for other even values of {{math|''n''}}, this is {{math|D{{sub|''n''}} / Z{{sub|2}}}}.

The [[cycle graph (group)|cycle graphs]] of dihedral groups consist of an ''n''-element cycle and ''n'' 2-element cycles. The dark vertex in the cycle graphs below of various dihedral groups represents the identity element, and the other vertices are the other elements of the group. A cycle consists of successive powers of either of the elements connected to the [[identity element]].
{| class=wikitable
|+ Cycle graphs
|-
! D<sub>1</sub> = [[Cyclic group|Z<sub>2</sub>]] || D<sub>2</sub> = Z<sub>2</sub><sup>2</sup> = [[Klein four-group|K<sub>4</sub>]] || D<sub>3</sub> || D<sub>4</sub> || D<sub>5</sub>
|-
| [[File:GroupDiagramMiniC2.svg|100px]] || [[File:GroupDiagramMiniD4.svg|100px]] || [[File:GroupDiagramMiniD6.svg|100px]] || [[File:GroupDiagramMiniD8.svg|100px]] || [[File:GroupDiagramMiniD10.svg|100px]]
|-
| [[File:GroupDiagramMiniD12.svg|100px]] || [[File:GroupDiagramMiniD14.svg|100px]] || [[File:GroupDiagramMiniD16.svg|100px]] || [[File:GroupDiagramMiniD18.png|100px]] || [[File:GroupDiagramMiniD20.png|100px]]
|-
! D{{sub|6}} {{=}} D{{sub|3}} × Z{{sub|2}} || D{{sub|7}} || D<sub>8</sub> || D<sub>9</sub> || D{{sub|10}} {{=}} D{{sub|5}} × Z{{sub|2}}
|}

{| class=wikitable style="text-align:center; vertical-align:top;"
|-
! [[Dihedral group of order 6|D<sub>3</sub>]] = [[Symmetric group|S]]<sub>3</sub>
! [[Dihedral group of order 8|D<sub>4</sub>]]
|-
| [[File:Symmetric group 3; cycle graph.svg|240px]]
| [[File:Dih4 cycle graph.svg|240px]]
|}