Dicyclic group

For each integer n > 1, the dicyclic group Dicn can be defined as the subgroup of the unit quaternions generated by More abstractly, one can define the dicyclic group Dicn as the group with the following presentation[3] Some things to note which follow from this definition: Thus, every element of Dicn can be uniquely written as amxl, where 0 ≤ m < 2n and l = 0 or 1.

The multiplication rules are given by It follows that Dicn has order 4n.

But the presentation of a dihedral group would have x2 = 1, instead of x2 = an; and this yields a different structure.

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 Dicn.

Indeed, the center consists solely of the identity element and x2.

If we add the relation x2 = 1 to the presentation of Dicn one obtains a presentation of the dihedral group Dihn, so the quotient group Dicn/ is isomorphic to Dihn.

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 Dihn.

Note that the dicyclic group does not contain any subgroup isomorphic to Dihn.

Let A be an abelian group, having a specific element y in A with order 2.

produces an ordinary triangle group, which in this case is the dihedral quotient