Dicyclic group
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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 di-cyclic).
Definition
For each integer n > 1 define the dicyclic group Dicn 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:
- x4 = 1
- x2ak = ak+n = akx2
- if j = ±1, then xjak = a-kxj.
- akx−1 = ak−nanx−1 = ak−nx2x−1 = ak−nx.
Thus, every element of Dicn can be uniquely written as akxj, 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^{k-m}x<math>
- <math>a^k x a^m x = a^{x-m+n}<math>
It follows that Dicn has order 4n. A specific representation of Dicn 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 Dicn is a non-abelian group of order 4n. (One doesn't consider "Dic1" as dicyclic).
Let A = <a> be the subgroup of Dicn generated by a. Then A is cyclic group of order 2n, so [Dicn:A] = 2. As a subgroup of index 2 it is automatically a normal subgroup. The quotient group Dicn/A is a cyclic group of order 2. Thus, Dicn is a cyclic extension of A.
Dicn 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 x2 = 1, instead of x2 = an; and this yields a different structure. In particular, Dicn 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 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 D2n, so the quotient group Dicn/<x2> is isomorphic to D2n.
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 the just the dihedral symmetry group D2n. 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 D2n.
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, x2 = y, and for all a in A, x-1ax = 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.