Definition:Dicyclic Group/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 $\map {\MM_\C} 2$:

$\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

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