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I know that the dihedral group is $$D_{2n} = \{1, r, r^2, \dotsc, r^{n-1}, s, sr, \dotsc, sr^n-1\}$$ where the $r^i$ are rotations and $s$ is a symmetry.

Now, what I want to know is what is the nature of $s$. It's a symmetry, but is it symmetry to the $y$ axis, to the $x$ axis, or to the origin? I thought it would be symmetric to the origin, but there's the following property $srs= r^{-1}$ but if $s$ is a symmetry according to the origin then I just get $srs=r$.

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  • $\begingroup$ $s$ actually stands for "Spiegelung" (maybe), which is German for "reflection". It's a reflection. $\endgroup$ – user399601 Apr 9 '17 at 17:48
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    $\begingroup$ It's a symmetry w.r.t. any of the $n$ axes of symmetry of the regular $n$-gon. $\endgroup$ – Bernard Apr 9 '17 at 17:55
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    $\begingroup$ $s$ is a reflection about any axis of symmetry. The choice of the axis will result in groups all isomorphic to one another. $\endgroup$ – Kaynex Apr 9 '17 at 17:56
  • $\begingroup$ You might be interested to read this carefull explanation: math.uconn.edu/~kconrad/blurbs/grouptheory/dihedral.pdf $\endgroup$ – Mathematician 42 Apr 9 '17 at 17:56
  • $\begingroup$ The relation srs=r can be rewritten as sr=rs. Thus that group is abelian. $\endgroup$ – i. m. soloveichik Apr 11 '17 at 11:38
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Geometrically, the Dihedral group $ D_{2n}$ is the group os symmetries of a regular polygon with $n$ sides.

For this planar figure you can find to types of symmetries: $n$ rotational symmetries (denoted by $r$) and reflection symmetries (denoted by $s$).

To describe this group algebraically, you pick some rotation of order $n$ and call it $r$, and some axis (not just $x$ or $y$) that divides the figure in two equal pieces and call the reflection through that axis $s$. The way you should thing about multiplication in this group is as composition of these two movements.

Just found this on youtube, I think it can help you solve your doubts: https://www.youtube.com/watch?v=rPh7EQPSaO4

Also: https://en.wikipedia.org/wiki/Dihedral_group

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