Dicyclic Group - Definition

Definition

For each integer n > 1, the dicyclic group Dicn can be defined as the subgroup of the unit quaternions generated by

\begin{align} a & = e^{i\pi/n} = \cos\frac{\pi}{n} + i\sin\frac{\pi}{n} \\ x & = j \end{align}

More abstractly, one can define the dicyclic group Dicn as any group having the presentation

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

It follows that Dicn has order 4n.

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.

Read more about this topic:  Dicyclic Group

Famous quotes containing the word definition:

    “Perhaps the best definition of progress would be the continuing efforts of men and women to narrow the gap between the convenience of the powers that be and the unwritten charter.”
    —Nadine Gordimer (b. 1923)

    “Was man made stupid to see his own stupidity?
    Is God by definition indifferent, beyond us all?
    Is the eternal truth man’s fighting soul
    Wherein the Beast ravens in its own avidity?”
    —Richard Eberhart (b. 1904)

    “I’m beginning to think that the proper definition of “Man” is “an animal that writes letters.””
    —Lewis Carroll [Charles Lutwidge Dodgson] (1832–1898)