Conjugacy classes for rotations of $D_{2n}$ It says in my notes that for a dihedral group $D_{2n}$, if $n$ is even, then each conjugacy class has at most $1$ element. It says that for the conjugacy class $C_{r^k}$ where $r^k$ is some reflection, the conjugacy class is $\{r^{\frac{n}{2} = k}\}$ (or just $\{r^{k}\}$).
I didn't follow this. Say we have $D_8$. Then $n = 4$, so $k = 2$. Then $r^2$ is just moving a vertex on a octagon over two vertices. But what about $r^1$, $r^3$, $r^5$, and $r^7$? $1$, $3$, $5$, and $7$ are not even numbers, so this says that they are not in any conjuacy classes since they are not of the form $r^2$, and the only conjuacy classes of $D_8$ is $\{r^{2}\}$. But we know that each element should belong to at least one conjugacy class, so how is this possible?

Something is certainly incorrect in your notes: a group is abelian if and only if every conjugacy class has size one, and the dihedral group $D_{2n}$ is not abelian for $n>2$. Possibly what is meant is that the center of $D_{2n}$ consists of at most one non-trivial element; and for $n$ even, this element is $r^{n/2}$.
