Dicyclic Group is a Group

Theorem

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

Corollary

The quaternion group $Q_4$, as it is an example of the dicyclic group with $4$ elements, is also a non-abelian group.

Proof

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

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

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

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