Properties:: [[quaternion group is nonabelian, but all its subgroups are normal]] Sufficiencies:: *[[Sufficiencies]]* Equivalences:: *[[Equivalences]]* ---- > [!definition] Definition. ([[quaternion group]]) > The [[subgroup]] $Q$ of $GL_{2}(\mathbb{C})$ generated by the [[matrix|matrices]] $a:=\begin{bmatrix} i & \\ & -i \end{bmatrix},b:= \begin{bmatrix} & -1 \\ 1 & \end{bmatrix}$ is called the **quaternion group**. Explicitly, $Q=\{ \v 1,a,a^{2},a^{3},b,ba,ba^{2} ,ba ^{3}\}, \text{ where}$$ \begin{align} \v 1 =& \begin{bmatrix} 1 & \\ & 1 \end{bmatrix}\\ a =&\begin{bmatrix}i & 0 \\ 0 & -i \end{bmatrix} \\ a^2 =& \begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix} \\ a^3 =& \begin{bmatrix} -i & 0 \\ 0 & i \end{bmatrix} \\ b =& \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} \\ ba = &\begin{bmatrix} 0 & i \\ i & 0 \end{bmatrix} \\ ba^2 =& \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \\ ba^3 =& \begin{bmatrix} 0 & -i \\ -i & 0 \end{bmatrix}\end{align}$ \ Actually, the elements $a$ and $b$ play completely symmetric roles, so it is more traditional to rename the elements of the group as follows: $\v i := \begin{bmatrix} i & \\ & -i \end{bmatrix}, \v j := \begin{bmatrix} & -1 \\ 1 & \end{bmatrix}, \v k:= \v i \v j.$ Then $Q=\{ \pm \v 1, \pm \v i ,\pm \v j, \pm \v k \}$ and we have $\v i ^{2}= \v j ^{2} =\v k^{2}=- \v 1, \v i \v j=\v k=-\v j \v i, \v j \v k = \v i= - \v k \v j, \v k \v i = \v j=-\v i \v k.$ ![[CleanShot 2023-09-17 at 18.07.55.jpg]] ^8a0ac7 ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```