# How to prove the following facts about Dihedral Groups

I am reading about Dihedral Groups and I have following questions:

• Elements of $$D_n$$ act as linear transformations of plane.

My thought:I know that $$D_n=\{\langle a,b\rangle :a^n=b^2=1,bab=a^{-1}\}$$ which comprises of rotations and reflections of the n-gon.

But then How to prove that rotations and reflections are linear transformations of plane?

• Matrices for elements of $$D_n$$ have the form: $$r_k$$=$$\begin{bmatrix} \cos{\frac{2\pi k}{n}}&-\sin{\frac{2\pi k}{n}}\\\sin{\frac{2\pi k}{n}}&\cos{\frac{2\pi k}{n}}\end{bmatrix}$$

which is obtained by rotation of a n-gon by $$\frac{2k\pi}{n}$$.

and $$s_k$$=$$\begin{bmatrix} \cos{\frac{2\pi k}{n}}&\sin{\frac{2\pi k}{n}}\\\sin{\frac{2\pi k}{n}}&-\cos{\frac{2\pi k}{n}}\end{bmatrix}$$

is a reflection about a line which makes an angle $$\frac{k\pi}{n}$$ with x-axis.

My thought:

I know that rotation matrix is given by $$\begin{bmatrix} \cos{\theta}&-\sin{\theta}\\\sin{\theta}&\cos{\theta}\end{bmatrix}$$.

But I dont know how a reflection matrix looks like? How can I prove that the elements of $$D_n$$ can be represented like this?

NOTE: The two lines have been adopted from Wikipedia. But I need a proof of these facts which I cant prove using my knowledge.

• – Gerry Myerson Nov 6 '18 at 11:44