Definition:Dicyclic Group
Definition
For even $n$, the dicyclic group $\Dic n$ of order $4 n$ is the group having the presentation:
- $\Dic n = \gen {a, b: a^{2 n} = e, b^2 = a^n, b^{-1} a b = a^{-1} }$
Quaternion Group
The dicyclic group $\Dic 2$ is known as the quaternion group.
The elements of $\Dic 2$ are:
- $\Dic 2 = \set {e, a, a^2, a^3, b, a b, a^2 b, a^3 b}$
Also denoted as
Some sources denote the dicyclic group $\Dic n$ as $Q_{2 n}$, referring to it as the generalized quaternion group:
- $Q_{2 n} = \gen {a, b: a^{2 n} = e, b^2 = a^n, b^{-1} a b = a^{-1} }$
Using this notation, it can be seen that the quaternion group is represented by:
- $Q_4 = \gen {a, b: a^4 = e, b^2 = a^2, b^{-1} a b = a^{-1} }$
Others have a different notation again:
- $Q_{4 n} = \gen {a, b: a^{2 n} = e, b^2 = a^n, b^{-1} a b = a^{-1} }$
Using this notation, it can be seen that the quaternion group is represented by:
- $Q_8 = \gen {a, b: a^4 = e, b^2 = a^2, b^{-1} a b = a^{-1} }$
Because of the potential ambiguity, it is recommended that $Q_{2 n}$ and $Q_{4 n}$ are not used, but that (except for the quaternion group itself, which $\mathsf{Pr} \infty \mathsf{fWiki}$ denotes $Q$) $\Dic n$ is used throughout.
Also see
- Results about dicyclic groups can be found here.
Linguistic Note
The word dicyclic in the term dicyclic group is parsed di-cyclic, that is, having two cycles.
Hence it is pronounced di-cy-clic.
Please do not say dicky-click.