Definition:Dicyclic Group

From ProofWiki
Jump to: navigation, search

Definition

For even $n$, the dicyclic group $\Dic n$ of order $4 n$ is the group having the presentation:

$\Dic n = \gen {x, y: x^{2 n} = e, y^2 = x^n, y^{-1} x y = x^{-1} }$


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 group $\Dic n$ as $Q_{2 n}$, referring to it as the generalized quaternion group:

$Q_{2 n} = \gen {x, y: x^{2n} = e, y^2 = x^n, y^{-1} x y = x^{-1} }$

Using this notation, it can be seen that the quaternion group is represented by:

$Q_4 = \gen {x, y: x^4 = e, y^2 = x^2, y^{-1} x y = x^{-1} }$


Others have a different notation again:

$Q_{4 n} = \gen {x, y: x^{2 n} = e, y^2 = x^n, y^{-1} x y = x^{-1} }$

Using this notation, it can be seen that the quaternion group is represented by:

$Q_8 = \gen {x, y: x^4 = e, y^2 = x^2, y^{-1} x y = x^{-1} }$


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.