Number of conjugacy classes of the reflection in $D_n$.

Question

Consider the conjugation action of $D_n$ on $D_n$. Prove that the number of conjugacy classes of the reflections are

$\begin{cases} 1 &\text{ if } n=\text{odd} \\ 2 &\text{ if } n=\text{even} \end{cases} $

I tried this: Let $σ$ be a reflection. And $ρ$ be the standard rotation of $D_n$.

$$ρ^l⋅σρ^k⋅ρ^{-l}=σρ^{k-2l}$$ $$σρ^l⋅σr^k⋅ρ^{-l}σ=σρ^{-k+2l}$$

If $n$ is even, it depends on $k$ if $-k+2l$ will stay even. But if $n$ is odd, then at some point $-k+2l=|D_n|$ and therefore you will also get the even elements. So independent of $k$ you will get all the elements. Is this the idea ?

Answer 1

Hint. The Orbit-Stabilizer theorem gives you that $[G:C_G(g)]$ is the size of the conjugacy class containing $g$. When $n$ is odd, a reflection $g$ commutes only with itself (why?), so $g$ has $[G:C_G(g)]=|G|/2$ elements, which are easily identified as the other reflections. Now, use this same technique to figure out the answer for the case of even $n$, keeping in mind that dihedral groups $D_{2n}$ have nontrivial centers when $n$ is even (why?).

Answer 2

A different approach: Draw a picture!

If $n$ is even, some reflection in $D_n$ are about lines connecting two opposite vertices, some reflections are bout lines connecting the midpoints of two opposite edges. The two types aren't related by conjugacy (see below). OTOH if $n$ is odd, all the symmetry axes connect a vertex to a midpoint of an edge.

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If $\sigma$ is the reflection about line $L$, and $\tau$ is any rotation of the plane around the origin, then $\tau\sigma\tau^{-1}$ is the reflection about the line $\tau(L)$: it maps points on the line $\tau(L)$ to themselves, preserves angles and distances, and has eigenvalue $-1$, so it is a reflection.