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Construct a symmetry group for the set of integers on the number line that generalizes the dihedral group to have a countably infinite, rather than finite size. Treat the integers as vertices.

What exactly would you need to construct this symmetry group? What properties other than the flip and the rotation would you need to show? I know that the flip would just be multiplication by $-1$ and that rotation counter-clockwise would be adding $1$ and rotation clockwise would be subtracting $1$.

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  • $\begingroup$ I think you have the right idea. There are the involutions $i_a:n\mapsto a-n$ that are the reflections, and the translations $t_d:n\mapsto n+d$. $\endgroup$ – Lubin Oct 14 '16 at 3:51
  • $\begingroup$ @Lubin I am sorry about my lack of knowledge of dihedral groups, but in the finite case, I can picture a rotation and flip fine. But I am having trouble what the meaning of a flip is in the infinite case, I am having trouble defining it. $\endgroup$ – ultrainstinct Oct 14 '16 at 4:55
  • $\begingroup$ If you have a definition or characterization of the infinite dihedral group, I think you’ll see that these generators satisfy. $\endgroup$ – Lubin Oct 14 '16 at 14:05
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When we generalize a concept from finite set to infinite set there could be many possible directions for generalizations. Your concept of rotation is plausible.

There are reflections in finite cases swapping two elements and keeping the others fixed. In the group generated by the two elements you describe there does not seem to be any simple swap. If such an element is desirable you should go for a different definition.

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  • $\begingroup$ So then in the case of the integers, do you need to flip the whole number line about the point 0 for a flip? I don't exactly understand the concept of the flip in the infinite case. $\endgroup$ – ultrainstinct Oct 14 '16 at 4:37
  • $\begingroup$ That is what you have already described by the function $n\mapsto -n$, this swaps every integer with its negative. Can we get the permutation that fixes everything except two integers? You have a real life problem requiring an infinite version, does the model require such a simple swap? The model should drive you towards the appropriate generalization. $\endgroup$ – P Vanchinathan Oct 14 '16 at 5:34
  • $\begingroup$ In my case, why would it be relevant to fix everything except the two integers? Does my reflection + rotation not construct a symmetry group on the whole countably infinite integer number line? $\endgroup$ – ultrainstinct Oct 14 '16 at 5:40
  • $\begingroup$ You misunderstood me. For you the (infinite) set of integers is a model (a finite set of integers is a model of vertices of a regular polygon) of some application in your domain that you know, not me. You have to take the call: does simple swap have significance, or arise naturally in your context? $\endgroup$ – P Vanchinathan Oct 14 '16 at 5:47

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