# Conjugate of elements in the Dihedral Group

I was trying to do the following from a past exam of my Rings and Groups' professor

Classify all conjugacy classes of the elements in the dihedral group $$D_n$$ = $$\{ 1,r,r^2, ... , r^{n-1} ,s ,rs ,r^2s , ... ,r^{n-1} \}$$ = $$\mathopen{<} r,s \ | \ r^n=1; s^2=1, (rs)^2=1 \mathopen{>}$$

A brief reminder that $$D_n$$ can be understood from a geometrical point of view, $$D_n$$ is the group of "symmetries" of a regular $$n$$-gon.

Given two elements $$x,a$$ in a group G, we define the conjugate of $$a$$ through $$x$$ as the element $$x^{-1}ax$$

We define the conjugacy class of an element $$a$$ in a group G as $$[a]=\{ x^{-1}ax \ | \ x \in G \}$$

More specifically, I'm trying to calculate the following :

1. The conjugate of a rotation $$r^k$$ through a rotation $$r^j$$ and through a symmetry $$r^js$$.
2. Determine the conjugacy class of $$r^k$$ in $$D_n$$. How many elements does it have?
3. The conjugate of a symmetry $$s$$ through a rotation $$r^j$$ and through a symmetry $$r^js$$.
4. Determine the conjugacy class of $$s$$ in $$D_n$$. How many elements does it have?

1. Evidently through $$r^j$$ we have, $$r^{-j}r^kr^j = r^{-j+k+j}=r^k$$.

Now through $$r^js$$ we get, $$(r^js)^{-1}r^k(r^js) = s^{-1}r^{-j}r^kr^js = s^{-1}r^ks$$.

I don't know how to classify this last one, and the ones from 3.

It should be sufficient to solve this to know that $$s^{-1}=s$$ which you can see from $$s^2=1$$, so $$s^2s^{-1}=s^{-1}$$, and since $$rsrs = 1$$ we have $$srs = r^{-1}$$ so $$sr = r^{-1}s$$ I'm not going to answer all of the questions, but this should be enough to do it yourself.