Prove the reflections of the dihedral group can be written in the form $r^hs$

Question

I saw in some exercise that the 2n elements of the dihedral group $D_n$ were written as $1 , r, r^2, ... , r^{n−1}, s, r s, r^2s, ... , r^{n−1}s$,

$r$ being the $2\pi /n$ counter-clockwise rotation and

$s$ the reflection about one axis of symmetry

How come the $ n$ reflections are written in the form $r^hs$?

Answer 1

It's easy to check using a presentation, say $\langle r,s|r^n,s^2,(rs)^2\rangle$, that the elements of the form $r^hs$ are all distinct from each other and from the rotations, and have order two.

Playing around with it a little bit, one can see that if $r$ corresponds to a rotation through $2π/n$, that the $r^hs$ are indeed the reflections.

Intuitively, the axis of reflection for $s$ just gets rotated successively.