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:

    “The physicians say, they are not materialists; but they are:MSpirit is matter reduced to an extreme thinness: O so thin!—But the definition of spiritual should be, that which is its own evidence. What notions do they attach to love! what to religion! One would not willingly pronounce these words in their hearing, and give them the occasion to profane them.”
    —Ralph Waldo Emerson (1803–1882)

    “It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possess—after many mysteries—what one loves.”
    —François, Duc De La Rochefoucauld (1613–1680)

    “According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animals—just as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
    —Ana Castillo (b. 1953)