Dicyclic Group - Binary Dihedral Group

Binary Dihedral Group

The dicyclic group is a binary polyhedral group — it is one of the classes of subgroups of the Pin group Pin₋(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_2n, the cyclic group C_n, and the dihedral group Dih_n of order 2n is illustrated in the diagram at right, and parallels the corresponding diagram for the Pin group.

There is a superficial resemblance between the dicyclic groups and dihedral groups; both are a sort of "mirroring" of an underlying cyclic group. But the presentation of a dihedral group would have x2 = 1, instead of x2 = an; and this yields a different structure. In particular, Dic_n is not a semidirect product of A and <x>, since A ∩ <x> is not trivial.

The dicyclic group has a unique involution (i.e. an element of order 2), namely x2 = an. Note that this element lies in the center of Dic_n. Indeed, the center consists solely of the identity element and x2. If we add the relation x2 = 1 to the presentation of Dic_n one obtains a presentation of the dihedral group Dih_2n, so the quotient group Dic_n/<x2> is isomorphic to Dih_n.

There is a natural 2-to-1 homomorphism from the group of unit quaternions to the 3-dimensional rotation group described at quaternions and spatial rotations. 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_n. 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_n.

The analogous pre-image construction, using Pin₊(2) instead of Pin₋(2), yields another dihedral group, Dih_2n, rather than a dicyclic group.