----- > [!proposition] Proposition. ([[dihedral group and linear isomorphisms of the plane]]) > The [[dihedral group]] $D_{n}$ is [[group isomorphism|isomorphic]] to the [[subgroup]] $H$ of the [[general linear group]] $GL_{2}(\mathbb{R})$ [[generating set of a group|generated by]] $O:=\begin{bmatrix} \cos \frac{2\pi}{n} & -\sin \frac{2\pi}{n} \\ \sin \frac{2\pi}{n} & \cos \frac{2\pi}{n} \end{bmatrix} \text{ and } R:= \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix},$ $\{ I_{2}, O, O^{2},\dots,O^{n-1}, RO, \dots , RO^{n-1} \}.$ ^dae82a > [!proof]- Proof. ([[dihedral group and linear isomorphisms of the plane]]) >Define the map $\phi:D_{n} \to H$ by > - $\phi(e)=\begin{bmatrix}1&0\\0&1\end{bmatrix}$; >- $\phi(x)=O$; >- $\phi(y)=R$; >- $\phi(x^{2}):=\phi(x)\phi(x)=O^{2}=\begin{bmatrix}\cos \frac{4\pi}{n} & \sin \frac{4\pi}{n} \\ \sin \frac{4\pi}{n} & -\cos \frac{4\pi}{n} \end{bmatrix}$; >- $\vdots$ >- $\phi(x^{n-1}):=\overbrace{\phi(x)\dots \phi(x)}^{n-1 \text{ times}}=O^{n-1}=\begin{bmatrix}\cos \frac{2\pi(n-1)}{n} & \sin \frac{2\pi(n-1)}{n} \\ \sin \frac{2\pi(n-1)}{n} & -\cos \frac{2\pi(n-1)}{n} \end{bmatrix}$; >- $\phi(yx):=\phi(y)\phi(x)=RO=\begin{bmatrix}\cos \frac{2\pi}{n} & \sin \frac{2\pi}{n} \\ \sin \frac{2\pi}{n} & -\cos \frac{2\pi}{n} \end{bmatrix}$; >- $\vdots$ >- $\phi(yx^{n-1}):=\phi(y)\overbrace{\phi(x)\dots \phi(x)}^{n-1 \text{ times}}=RO^{n-1}=\begin{bmatrix}\cos \frac{2\pi(n-1)}{n} & \sin \frac{2\pi(n-1)}{n} \\ \sin \frac{2\pi(n-1)}{n} & -\cos \frac{2\pi(n-1)}{n} \end{bmatrix}$. \ By construction, $\phi$ is a [[group homomorphism|homomorphism]]. It is an [[injection]] according to the explicit map given above, and a [[surjection]] as an [[injection]] between finite sets of equal [[cardinality]]. ----- #### ---- #### 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 > ```