# Dihedral Group and Rotation of the Plane

## Problem 52

Let $n$ be a positive integer. Let $D_{2n}$ be the dihedral group of order $2n$. Using the generators and the relations, the dihedral group $D_{2n}$ is given by
$D_{2n}=\langle r,s \mid r^n=s^2=1, sr=r^{-1}s\rangle.$ Put $\theta=2 \pi/n$.

(a) Prove that the matrix $\begin{bmatrix} \cos \theta & -\sin \theta\\ \sin \theta& \cos \theta \end{bmatrix}$ is the matrix representation of the linear transformation $T$ which rotates the $x$-$y$ plane about the origin in a counterclockwise direction by $\theta$ radians.

(b) Let $\GL_2(\R)$ be the group of all $2 \times 2$ invertible matrices with real entries. Show that the map $\rho: D_{2n} \to \GL_2(\R)$ defined on the generators by
$\rho(r)=\begin{bmatrix} \cos \theta & -\sin \theta\\ \sin \theta& \cos \theta \end{bmatrix} \text{ and } \rho(s)=\begin{bmatrix} 0 & 1\\ 1& 0 \end{bmatrix}$ extends to a homomorphism of $D_{2n}$ into $\GL_2(\R)$.

(c) Determine whether the homomorphism $\rho$ in part (b) is injective and/or surjective.

## Hint.

1. For (a), consider the unit vectors of the plane and consider where do the unit vector go by the linear transformation $T$.
2. Show that $\rho(r)$ and $\rho(s)$ satisfy the same relations as $D_{2n}. 3. Consider the determinant. ## Proof. ### (a) The matrix representation of the linear transformation$T$Let$\mathbf{e}_1, \mathbf{e}_2$be the standard basis of the plane$\R^2$. That is $\mathbf{e}_1=\begin{bmatrix} 1 \\ 0 \end{bmatrix} \text{ and } \mathbf{e}_2=\begin{bmatrix} 0 \\ 1 \end{bmatrix}.$ Then by the$\theta$rotation$\mathbf{e}_1$moves to the point$\begin{bmatrix}
\cos \theta \\
\sin \theta
\end{bmatrix}$and$\mathbf{e}_2$moves to the point$\begin{bmatrix}
-\sin \theta \\
\cos \theta
\end{bmatrix}$. Therefore the matrix representation of$T$is the matrix$\begin{bmatrix}
\cos \theta & -\sin \theta\\
\sin \theta& \cos \theta

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