Show that $D_{2n}$ has two conjugacy classes of reflections if $n$ is even, but only one if $n$ is odd 
Show that $D_{2n}$ has two conjugacy classes if $n$ is even, but only one if $n$ is odd.

My questions:


*

*What is meant by 'two conjugacy classes of reflections'

*How is this question related to whether $n$ is even or odd
I am thinking about orbit-stabiliser theorem, and find that, if $D_{2n}=<r,s|r^n=e=s^2,sr=r^{-1}s>$, $R_n=<r>$ and $K \leq R_n$, then$$|\text{ccl}(s)|=n$$ when $n$ is odd, and $$|\text{ccl}(s)|={n \over 2}$$ when $n$ is even.
However I cannot see how is this result related to the question, can someone please give me a hint?
 A: You've done almost all the work you need to do, it's just a matter of relating it to the goal.
You know that, no matter whether $n$ is even or odd, $D_{2n}$ has exactly $n$ reflections. You should also be able to show that the only elements conjugate to a reflection are themselves reflections. So if we let $s$ be a fixed but arbitrary reflection and $S$ the set of reflections, make sure you mention/show that $\operatorname{ccl}(s) \subseteq S$.
When $n$ is odd, you've shown correctly that $\lvert \operatorname{ccl}(s) \rvert = n$ while $\operatorname{ccl}(s)$ must consist entirely of reflections, of which there are only $n$, and we must have $\operatorname{ccl}(s) = S$.
A very similar argument happens when $n$ is even. Still $\operatorname{ccl(s)} \subseteq S$ for any reflection $s \in S$, and since each reflection is conjugate to exactly $\frac{n}{2}$ others, $S$ must be the disjoint union $S = \operatorname{ccl}(s) \sqcup \operatorname{ccl}(t)$ for two rotations $s, t \in S$, purely for counting/elementary set theoretical reasons.
And it helps to keep the geometry in mind.

It's pretty clear that, geometrically, there are "two kinds" of reflections for a square, and the cool thing is that this is reflected in the group structure of $D_{8}$. But if you think about, say, a pentagon, every reflection axis goes from a vertex to the midpoint of the opposite edge, which causes all reflections to be conjugate in $D_{10}$.
