Editing Quaternion group (section)

== Compared to dihedral group ==

The quaternion group Q<sub>8</sub> has the same order as the [[dihedral group]] [[Examples of groups#The symmetry group of a square: dihedral group of order 8|D<sub>4</sub>]], but a different structure, as shown by their Cayley and cycle graphs:

{| class="wikitable" style="width: 500px; text-align: center;"
!
! Q<sub>8</sub>
! [[dihedral group of order 8|D<sub>4</sub>]]
|-
! [[Cayley graph]]
|style="vertical-align:top;"| [[Image:Cayley graph Q8.svg|200px]]<br/>Red arrows connect ''g''<span style="color:red;">→</span>''gi'', green connect ''g''<span style="color:lightgreen;">→</span>''gj''.
|style="vertical-align:top;"| [[File:Dih 4 Cayley Graph; generators a, b.svg|220px]]
|-
! [[Cycle graph (algebra)|Cycle graph]]
| [[File:GroupDiagramQ8.svg|120px]]
| [[File:Dih4 cycle graph.svg|120px]]
|}

In the diagrams for D<sub>4</sub>, the group elements are marked with their action on a letter F in the defining representation '''R'''<sup>2</sup>. The same cannot be done for Q<sub>8</sub>, since it has no faithful representation in '''R'''<sup>2</sup> or '''R'''<sup>3</sup>. D<sub>4</sub> can be realized as a subset of the [[split-quaternion]]s in the same way that Q<sub>8</sub> can be viewed as a subset of the quaternions.