## All the Conjugacy Classes of the Dihedral Group $D_8$ of Order 8

## Problem 54

Determine all the conjugacy classes of the dihedral group

\[D_{8}=\langle r,s \mid r^4=s^2=1, sr=r^{-1}s\rangle\]
of order $8$.

Determine all the conjugacy classes of the dihedral group

\[D_{8}=\langle r,s \mid r^4=s^2=1, sr=r^{-1}s\rangle\]
of order $8$.

Let $D_8$ be the dihedral group of order $8$.

Using the generators and relations, we have

\[D_{8}=\langle r,s \mid r^4=s^2=1, sr=r^{-1}s\rangle.\]

**(a)** Let $A$ be the subgroup of $D_8$ generated by $r$, that is, $A=\{1,r,r^2,r^3\}$.

Prove that the centralizer $C_{D_8}(A)=A$.

**(b)** Show that the normalizer $N_{D_8}(A)=D_8$.

**(c) **Show that the center $Z(D_8)=\langle r^2 \rangle=\{1,r^2\}$, the subgroup generated by $r^2$.

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.