how to list elements of a dihedral group $D_n$

Question

A formula has been posted here (Is there a general formula for finding all subgroups of dihedral groups?). However, I'm unfamiliar of the notation used. With the rule stated on the link, how do I list the elements of a given dihedral group $D_n$. An example would be appreciated. Thanks!

Answer 1

(Refer https://en.wikipedia.org/wiki/Dihedral_group)
The group presentation of a dihedral group $D_n$ is $$\langle r,s:r^n=1,s^2=1,sr=r^{-1}s\rangle$$ Since $D_n$ is generated by $r$ and $s$, every element will be of the form $$r^{\alpha_1}s^{\beta_1}\dots r^{\alpha_k}s^{\beta_k}$$ where $0\leq\alpha_i<n, 0\leq\beta_i<2$ and $k\geq 1$.
Since we can always swap the position of $s$ and $r$ by using the relation $sr=r^{-1}s$, we can always write the element in the form $$r^\alpha s^\beta$$ where $0\leq\alpha_i<n, 0\leq\beta_i<2$.

Hence $D_n$ consists of $1,r,r^2,\dots,r^{n-1},s,rs,\dots,r^{n-1}s$. But to complete the proof, you still have to show that all the elements in the list are distinct. This will be left as an exercise for you.

Answer 2

You can take a rotation through $2π/n$, call it $\rho$. And a reflection $\sigma$. You can get all the other rotations as powers of $\rho$. And the other reflections by application of $\rho$ successively to $\sigma$. That's a reflection followed by a rotation is another reflection.

So we get $D_{2n}=\{e,\rho,\rho^2,\dots,\rho^{n-1},\sigma,\rho\sigma,\dots,\rho^{n-1}\sigma\}$.