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Here is a result about dihedral groups. $rs = sr ^{-1}$, where $r$ is a rotation of $\frac{2 \pi}{n}$ radians and $s$ is a reflection about the line of symmetry from vertex $i$ and the origin. This is quite understable, and I can basically prove this using my intuition, but I cannot reach a rigorous proof. I also thought that as this result is familiar, so I searched all the questions in MSE but I did not find any. So I posted this question. Can you please help?

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  • $\begingroup$ Don't ever think you can vandalize your questions and get away with it. $\endgroup$ – user21820 Apr 3 at 3:03
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Any reflection is its own inverse. Since $rs$ is a reflection we have $(rs)^{-1}=rs$. On the other hand we also know that $(rs)^{-1}=s^{-1}r^{-1}=sr^{-1}$. ($s=s^{-1}$ because $s$ is a reflection as well). Since the inverse is unique we conclude that $rs=sr^{-1}$.

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