Can a dihedral group $D_2$ be a set of transformations of a line segment? If it can then what are the reflections?

No matter how I think about reflecting it I always get either the identity or the result of one rotation.

Shouldn't the two reflections be something distinct to the identity and one rotation for it to be considered a dihedral group?

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If you want $D_2$ to be the set of symmetries of a line segment (a "regular $2$-gon"), then yes, it can be. But you have to take into account that flipping it over (the operation that has order $2$ in any dihedral group) is different from rotating it $180^\circ$ in the plane. So to truly generalize the case of $D_n$ being the group of symmetries of the regular $n$-gon, you have to see it as the set of symmetries of a line segment with an overside and an underside.

  • $\begingroup$ Precisely what I want to know is how is flipping it over different than rotating? If it is a vertical line segment and point $A$ is above and point $B$ is below then flipping or rotating would put the $B$ above, and $A$ below. $\endgroup$ – Michael Munta Sep 4 at 11:13
  • $\begingroup$ @MichaelMunta If you just draw a standard line segment, then the operations aren't different. But then $D_2$ isn't the symmetry group either, $C_2$ is. If you want $D_2$ to be the symmetry group, then your line segment has to be green on the overside (the side you see initially), and red on the underside (the side you see after flipping it). Or something similar. This is precisely to make the flipping and the rotation distinct operations. $\endgroup$ – Arthur Sep 4 at 11:16
  • $\begingroup$ So in this case I can not think about the different symmetries as permutations of vertices? Like, for example, in triangle we have 6 different symmetries which correspond to 6 permutations of the set of vertices. $\endgroup$ – Michael Munta Sep 4 at 11:23
  • $\begingroup$ @MichaelMunta You can see yourself that that isn't enough. There are only two permutations on two vertices, but $D_2$ has four elements. The closest you can get is the group of symmetries on a rectangle. $\endgroup$ – Arthur Sep 4 at 11:35

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