Dicyclic Group is Non-Abelian Group

Theorem
The dicyclic group $\Dic n$ is a non-abelian group on two generators.

Proof
The dicyclic group $\Dic n$ is defined as follows:

First it is to be demonstrated that $\Dic n$ is a group.

First we deduce the following:

$(1): \quad b^4 = e$:

$(2): \quad b^2 a^k = a^{k + n} = a^k b^2$:

$(3): \quad j = \pm 1 \implies b^j a^k = a^{-k} b^j$

Thus, every element of $\Dic n$ can be uniquely written as $a^k b^j$, where $0 \le k < 2 n$ and $j \in \set {0, 1}$.

The definition of the group product gives:


 * $a^k a^m = a^{k + m}$


 * $a^k a^m b = a^{k + m} b$


 * $a^k b a^m = a^{k - m} b$


 * $a^k b a^m x = a^{k - m + n}$

Taking the group axioms in turn:

Let $x, y \in \Dic n$.

Thus $\Dic n$ is closed.

Thus $\Dic n$ is associative.

Thus $e$ is the identity element of $\Dic n$.

We have that $e$ is the identity element of $\Dic n$.

Thus every element $...$ of $\Dic n$ has an inverse $...$.

All the group axioms are thus seen to be fulfilled, and so $...$ is a group.

Generators
The generator of $\Dic n$ is seen to be $\set {a, b}$.