Definition:Dicyclic Group/Quaternion Group
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:
$\quad \begin {array} {r|rrrrrrrr} & e & a & a^2 & a^3 & b & a b & a^2 b & a^3 b \\ \hline e & e & a & a^2 & a^3 & b & a b & a^2 b & a^3 b \\ a & a & a^2 & a^3 & e & a b & a^2 b & a^3 b & b \\ a^2 & a^2 & a^3 & e & a & a^2 b & a^3 b & b & a b \\ a^3 & a^3 & e & a & a^2 & a^3 b & b & a b & a^2 b \\ b & b & a^3 b & a^2 b & a b & a^2 & a & e & a^3 \\ a b & a b & b & a^3 b & a^2 b & a^3 & a^2 & a & e \\ a^2 b & a^2 b & a b & b & a^3 b & e & a^3 & a^2 & a \\ a^3 b & a^3 b & a^2 b & a b & b & a & e & a^3 & a^2 \end {array}$
Quaternion Group defined by Complex Matrices
Let $\mathbf 1, \mathbf i, \mathbf j, \mathbf k$ denote the following four elements of the matrix space $\map {\MM_\C} 2$:
- $\mathbf 1 = \begin {bmatrix} 1 & 0 \\ 0 & 1 \end {bmatrix}
\qquad \mathbf i = \begin {bmatrix} i & 0 \\ 0 & -i \end {bmatrix} \qquad \mathbf j = \begin {bmatrix} 0 & 1 \\ -1 & 0 \end {bmatrix} \qquad \mathbf k = \begin {bmatrix} 0 & i \\ i & 0 \end {bmatrix}$
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:
Quaternion Group defined by Matrices of Order $4$
Let $\mathbf I, \mathbf J, \mathbf K, \mathbf L$ denote the following four elements of the matrix space $\map {\MM_\Z} 4$:
\(\ds \mathbf I\) | \(=\) | \(\ds \begin {bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end {bmatrix}\) | ||||||||||||
\(\ds \mathbf J\) | \(=\) | \(\ds \begin {bmatrix} 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \end {bmatrix}\) | ||||||||||||
\(\ds \mathbf K\) | \(=\) | \(\ds \begin {bmatrix} 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & -1 & 0 \end {bmatrix}\) | ||||||||||||
\(\ds \mathbf L\) | \(=\) | \(\ds \begin {bmatrix} 0 & 0 & 0 & -1 \\ 0 & 0 & 1 & 0 \\ 0 & -1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end {bmatrix}\) |
where $\Z$ is the set of integers.
The set:
- $\Dic 2 = \set {\mathbf I, -\mathbf I, \mathbf J, -\mathbf J, \mathbf K, -\mathbf K, \mathbf L, -\mathbf L}$
under the operation of conventional matrix multiplication, forms the quaternion group.
This can be generated by the $2$ elements $\mathbf J$ and $\mathbf K$.
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.
Sources
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: Conjugacy, Normal Subgroups, and Quotient Groups: $\S 46 \iota$
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): quaternion group
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): quaternion group