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?

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    $\begingroup$ Let's say that the reflection $r$ is the permutation sending $1, 2, \ldots, n$ to $1, n, n-1, \ldots, 2$, so that it satisfies $r\left(i\right) \equiv 2-i \mod n$. You want to prove that the $2n$ maps $r^i,\ sr^i$ for $i \in \left\{0,1,\ldots,n-1\right\}$ are all distinct. Hint: Look at the images of $1$ and $n$ under each of these maps; these alone should suffice to tell them apart. (Assuming, of course, that $n > 2$; otherwise the claim is false for permutations.) $\endgroup$ – darij grinberg Nov 5 '18 at 17:42
  • $\begingroup$ @darijgrinberg Thanks for reply, you mean $\sigma(i)=2-i\pmod{n}$? $\endgroup$ – Taxicab Nov 5 '18 at 17:53
  • $\begingroup$ I don't want to call my $r$ $\sigma$, since you already use that letter for the $s$. Also, if your notation is a congruence modulo $n$, then we mean the same thing. $\endgroup$ – darij grinberg Nov 5 '18 at 17:54
  • $\begingroup$ Ok, can you give some more hint? how I can use the congruence mod n in composition here?(as $r^i$ means composing $r$, $i$ times, same for $sr^i$) that's the confusion! $\endgroup$ – Taxicab Nov 5 '18 at 17:57
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    $\begingroup$ Each $k \in \left\{1,2,\ldots,n\right\}$ satisfies $r\left(k\right) \equiv k+1\mod n$ and thus $r^i\left(k\right) \equiv k+i\mod n$ (prove this by induction on $i$) and thus $\left(sr^i\right)\left(k\right) \equiv 2-k-i\mod n$. Now draw your conclusions. $\endgroup$ – darij grinberg Nov 5 '18 at 18:01

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