Symmetry group on integers

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 $1$.

• I think you have the right idea. There are the involutions $i_a:n\mapsto a-n$ that are the reflections, and the translations $t_d:n\mapsto n+d$. – Lubin Oct 14 '16 at 3:51
• @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. – ultrainstinct Oct 14 '16 at 4:55
• If you have a definition or characterization of the infinite dihedral group, I think you’ll see that these generators satisfy. – Lubin Oct 14 '16 at 14:05

• 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. – P Vanchinathan Oct 14 '16 at 5:34