# Definition:Dicyclic Group

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## 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**.