Editing Dicyclic group (section)

==Binary dihedral group==
[[Image:Dicyclic-commutative-diagram.svg|right|400px]]

The dicyclic group is a [[binary polyhedral group]] — it is one of the classes of subgroups of the [[Pin group]] Pin<sub>−</sub>(2), which is a subgroup of the [[Spin group]] Spin(3) — and in this context is known as the '''binary dihedral group'''.

The connection with the [[binary cyclic group]] ''C''<sub>2''n''</sub>, the cyclic group ''C''<sub>''n''</sub>, and the [[dihedral group]] Dih<sub>''n''</sub> of order 2''n'' is illustrated in the diagram at right, and parallels the corresponding diagram for the Pin group. Coxeter writes the ''binary dihedral group'' as ⟨2,2,''n''⟩ and ''binary cyclic group'' with angle-brackets, ⟨''n''⟩.

There is a superficial resemblance between the dicyclic groups and [[dihedral group]]s; both are a sort of "mirroring" of an underlying cyclic group. But the presentation of a dihedral group would have ''x''<sup>2</sup> = 1, instead of ''x''<sup>2</sup> = ''a''<sup>''n''</sup>; and this yields a different structure. In particular, Dic<sub>''n''</sub> is not a [[semidirect product]] of ''A'' and {{angbr|''x''}}, since ''A''&nbsp;∩&nbsp;{{angbr|''x''}} is not trivial.

The dicyclic group has a unique [[involution (mathematics)#Group theory|involution]] (i.e. an element of order 2), namely ''x''<sup>2</sup> = ''a''<sup>''n''</sup>. Note that this element lies in the [[center of a group|center]] of Dic<sub>''n''</sub>. Indeed, the center consists solely of the identity element and ''x''<sup>2</sup>. If we add the relation ''x''<sup>2</sup> = 1 to the presentation of Dic<sub>''n''</sub> one obtains a presentation of the [[dihedral group]] Dih<sub>''n''</sub>, so the quotient group Dic<sub>''n''</sub>/<''x''<sup>2</sup>> is isomorphic to Dih<sub>''n''</sub>.

There is a natural 2-to-1 [[homomorphism]] from the group of unit quaternions to the 3-dimensional [[rotation group SO(3)|rotation group]] described at [[quaternions and spatial rotation]]s. Since the dicyclic group can be embedded inside the unit quaternions one can ask what the image of it is under this homomorphism. The answer is just the dihedral symmetry group Dih<sub>''n''</sub>. For this reason the dicyclic group is also known as the '''binary dihedral group'''. Note that the dicyclic group does not contain any subgroup isomorphic to Dih<sub>''n''</sub>.

The analogous pre-image construction, using Pin<sub>+</sub>(2) instead of Pin<sub>−</sub>(2), yields another dihedral group, Dih<sub>2''n''</sub>, rather than a dicyclic group.