Definition:Quaternion Group

Definition

The dicyclic group $Dic_2$ is known as the quaternion group.

The elements of $Dic_2$ are:

$Dic_2 = \left\{{e, a, a^2, a^3, b, a b, a^2 b, a^3 b}\right\}$

Its group presentation is given by:

$Dic_2 = \left \langle {a, b: a^4 = e, b^2 = a^2, a b a = b}\right \rangle$

Its Cayley table is given by:

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

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

Quaternion Group defined by Complex Matrices

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

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

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

The set:

$Q_4 = \left\{{\mathbf 1, -\mathbf 1, \mathbf i, -\mathbf i, \mathbf j, -\mathbf j, \mathbf k, -\mathbf k}\right\}$

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

$\begin{array}{c|cccccccc} & \mathbf 1 & \mathbf i & -\mathbf 1 & -\mathbf i & \mathbf j & \mathbf k & -\mathbf j & -\mathbf k \\ \hline \mathbf 1 & \mathbf 1 & \mathbf i & -\mathbf 1 & -\mathbf i & \mathbf j & \mathbf k & -\mathbf j & -\mathbf k \\ \mathbf i & \mathbf i & -\mathbf 1 & -\mathbf i & \mathbf 1 & \mathbf k & -\mathbf j & -\mathbf k & \mathbf j \\ -\mathbf 1 & -\mathbf 1 & -\mathbf i & \mathbf 1 & \mathbf i & -\mathbf j & -\mathbf k & \mathbf j & \mathbf k \\ -\mathbf i & -\mathbf i & \mathbf 1 & \mathbf i & -\mathbf 1 & -\mathbf k & \mathbf j & \mathbf k & -\mathbf j \\ \mathbf j & \mathbf j & -\mathbf k & -\mathbf j & \mathbf k & -\mathbf 1 & \mathbf i & \mathbf 1 & -\mathbf i \\ \mathbf k & \mathbf k & \mathbf j & -\mathbf k & -\mathbf j & -\mathbf i & -\mathbf 1 & \mathbf i & \mathbf 1 \\ -\mathbf j & -\mathbf j & \mathbf k & \mathbf j & -\mathbf k & \mathbf 1 & -\mathbf i & -\mathbf 1 & \mathbf i \\ -\mathbf k & -\mathbf k & -\mathbf j & \mathbf k & \mathbf j & \mathbf i & \mathbf 1 & -\mathbf i & -\mathbf 1 \end{array}$

Also see

The Quaternion Group is Hamiltonian.

Sources

• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 46 \iota$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): Exercise $6.14$