Definition:Quaternion Group

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Definition

The dicyclic group $\Dic 2$ is known as the quaternion group.

The elements of $\Dic 2$ are:

$\Dic 2 = \set {e, a, a^2, a^3, b, a b, a^2 b, a^3 b}$

Group Presentation

Its group presentation is given by:

$\Dic 2 = \gen {a, b: a^4 = e, b^2 = a^2, a b a = b}$

Cayley Table

Its Cayley table is given by:

$\begin{array}{r|rrrrrrrr} & e & a & a^2 & a^3 & b & a b & a^2 b & a^3 b \\ \hline e & e & a & a^2 & a^3 & b & a b & a^2 b & a^3 b \\ a & a & a^2 & a^3 & e & a b & a^2 b & a^3 b & b \\ a^2 & a^2 & a^3 & e & a & a^2 b & a^3 b & b & a b \\ a^3 & a^3 & e & a & a^2 & a^3 b & b & a b & a^2 b \\ b & b & a^3 b & a^2 b & a b & a^2 & a & e & a^3 \\ a b & a b & b & a^3 b & a^2 b & a^3 & a^2 & a & e \\ a^2 b & a^2 b & a b & b & a^3 b & e & a^3 & a^2 & a \\ a^3 b & a^3 b & a^2 b & a b & b & a & e & a^3 & a^2 \end{array}$

Quaternion Group defined by Complex Matrices

Let $\mathbf 1, \mathbf i, \mathbf j, \mathbf k$ denote the following four elements of the matrix space $\map {\mathcal M_\C} 2$:

$\mathbf 1 = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \qquad \mathbf i = \begin{bmatrix} i & 0 \\ 0 & -i \end{bmatrix} \qquad \mathbf j = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \qquad \mathbf k = \begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix}$

where $\C$ is the set of complex numbers.

The set:

$\Dic 2 = \set {\mathbf 1, -\mathbf 1, \mathbf i, -\mathbf i, \mathbf j, -\mathbf j, \mathbf k, -\mathbf k}$

under the operation of conventional matrix multiplication, forms the quaternion group:

Subgroups

The subsets of $Q$ which form subgroups of $Q$ are:

 $\ds$  $\ds Q$ $\ds$  $\ds \set e$ $\ds$  $\ds \set {e, a^2}$ $\ds$  $\ds \set {e, a, a^2, a^3}$ $\ds$  $\ds \set {e, b, a^2, a^2 b}$ $\ds$  $\ds \set {e, a b, a^2, a^3 b}$

From Quaternion Group is Hamiltonian we have that all of these subgroups of $Q$ are normal.

Also known as

Many sources (including this website) tend to refer to this group merely as $Q$.

Other sources use $Q_4$.

Also see

• Results about the quaternion group can be found here.