I am self studying Dihedral groups from Dummit and Foote abstract algebra book. It is given to prove:
$(1)$ $s\neq r^i$,$(2)~sr^i\neq sr^j,0\le i,j<n$ where $r $ is rotation by $2\pi /n$ angle clockwise and $s$ is reflection about the line passing through vertex $1$ and the center of the $n$-gon.
I think, geometrically it will be hard to show these and I am trying to treat rotations and reflections as permutations. Like, rotation can be treated as $\sigma(i)=i+1\pmod{n}$. But not getting how I can show these. Can anyone help to work with this idea please?