Dicyclic group

In group theory, a dicyclic group is a member of a class of groups which are formed by an extension of a group (generally a cyclic group) by a cyclic group of order 2 (the latter giving the name dicyclic).
Definition
For each integer n > 1 define the dicyclic group Dic_{n} as the group having the presentation
 <math>\mbox{Dic}_n = \langle a,x \mid a^{2n} = 1, x^2 = a^n, x^{1}ax = a^{1}\rangle.<math>
Some things to note which follow from this definition:
 x^{4} = 1
 x^{2}a^{k} = a^{k+n} = a^{k}x^{2}
 if j = ±1, then x^{j}a^{k} = a^{k}x^{j}.
 a^{k}x^{−1} = a^{k−n}a^{n}x^{−1} = a^{k−n}x^{2}x^{−1} = a^{k−n}x.
Thus, every element of Dic_{n} can be uniquely written as a^{k}x^{j}, where 0 ≤ k < 2n and j = 0 or 1. The multiplication rules are given by
 <math>a^k a^m = a^{k+m}<math>
 <math>a^k a^m x = a^{k+m}x<math>
 <math>a^k x a^m = a^{km}x<math>
 <math>a^k x a^m x = a^{xm+n}<math>
It follows that Dic_{n} has order 4n. A specific representation of Dic_{n} can be found using unit quaternions. Let
 <math>a = e^{i\pi/n} = \cos\frac{\pi}{n} + i\sin\frac{\pi}{n}<math>
 <math>x = j\,<math>
It is easy to see that a and x satisfy the relations for the dicyclic group.
When n = 2, the dicyclic group is isomorphic to the quaternion group Q. More generally, when n is a power of 2, the dicyclic group is isomorphic to the generalized quaternion group.
Properties
For each n > 1, the dicyclic group Dic_{n} is a nonabelian group of order 4n. (One doesn't consider "Dic_{1}" as dicyclic).
Let A = <a> be the subgroup of Dic_{n} generated by a. Then A is cyclic group of order 2n, so [Dic_{n}:A] = 2. As a subgroup of index 2 it is automatically a normal subgroup. The quotient group Dic_{n}/A is a cyclic group of order 2. Thus, Dic_{n} is a cyclic extension of A.
Dic_{n} is solvable; note that A is normal, and being abelian, is itself solvable.
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 x^{2} = 1, instead of x^{2} = a^{n}; 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 an unique involution (i.e. an element of order 2), namely x^{2} = a^{n}. Note that this element lies in the center of Dic_{n}. Indeed, the center consists solely of the identity element and x^{2}. If we add the relation x^{2} = 1 to the presentation of Dic_{n} one obtains a presentation of the dihedral group D_{2n}, so the quotient group Dic_{n}/<x^{2}> is isomorphic to D_{2n}.
There is a natural 2to1 homomorphism from the group of unit quaternions to the 3dimensional 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 the just the dihedral symmetry group D_{2n}. 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 D_{2n}.
Generalizations
Let A be an abelian group, having a specific element y in A with order 2. A group G is called a generalized dicyclic group, written as Dic(A, y), if it is generated by A and an additional element x, and in addition we have that [G:A] = 2, x^{2} = y, and for all a in A, x^{1}ax = a^{−1}.
Since for a cyclic group of even order, there is always a unique element of order 2, we can see that dicyclic groups are just a specific type of generalized dicyclic group.
Generalized dicyclic groups, in turn, are examples of cyclic extensions.