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

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