Dicyclic Group is Non-Abelian Group

Theorem
The dicyclic group is a non-abelian group on two generators.

Proof
From the definition of the group, all of these statements follow:


 * $$y^4 = 1 \ $$
 * $$y^2x^k = x^{k+n} = x^ky^2 \ $$
 * $$j = \pm 1 \Longrightarrow y^jx^k = x^{-k}y^j \ $$
 * $$y^ky^{-1} = x^{k-n}y^ny^{-1} = x^{k-n}y^2y^{-1} = x^{k-n}y \ $$

Thus, every element of $$Q_n \ $$ can be uniquely written as $$x^k y^j \ $$, where $$0 \leq k < 2n \ $$ and $$j \in \left\{{0,1}\right\} \ $$.

The multiplication rules are given by
 * $$a^k a^m = a^{k+m} \ $$
 * $$a^k a^m x = a^{k+m}x \ $$
 * $$a^k x a^m = a^{k-m}x \ $$
 * $$a^k x a^m x = a^{k-m+n} \ $$