Dicyclic group

In group theory, a dicyclic group (notation Dicn or Q4n, ) is a particular kind of non-abelian group of order 4n (n > 1). It is an extension of the cyclic group of order 2 by a cyclic group of order 2n, giving the name di-cyclic. In the notation of exact sequences of groups, this extension can be expressed as:

:<math>1 \to C_{2n} \to \mbox{Dic}_n \to C_2 \to 1. \, </math>

More generally, given any finite abelian group with an order-2 element, one can define a dicyclic group.

Definition

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

:<math>\begin{align}

a & = e^\frac{i\pi}{n} = \cos\frac{\pi}{n} + i\sin\frac{\pi}{n} \\

x & = j

\end{align}</math>

More abstractly, one can define the dicyclic group Dicn as the group with the following presentation

:<math>\operatorname{Dic}_n = \left\langle a, x \mid a^{2n} = 1,\ x^2 = a^n,\ x^{-1}ax = a^{-1}\right\rangle.\,\!</math>

Some things to note which follow from this definition:

<math> x^4 = 1 </math>
<math> x^2 a^m = a^{m+n} = a^m x^2 </math>
if <math> l = \pm 1 </math>, then <math> x^l a^m = a^{-m} x^l </math>
<math> a^m x^{-1}= a^{m-n} a^n x^{-1}= a^{m-n} x^2 x^{-1}= a^{m-n} x </math>

Thus, every element of Dicn can be uniquely written as , where 0 ≤ m < 2n and l = 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^{k-m+n}</math>

It follows that Dicn has order 4n.