Is there a presentation for the symmetries of the circle?

Question

If we take the group that is the real points on the unit circle on the plane under rotations and reflections for symmetry. Is this group $D_\infty $? Can it be generated by $\{ r,s\}$, and if so does it have a presentation similar to $D_{2n}$ for n finite?

Answer 1

Any finitely (or countably) generated group would only be countable. The group is isomorphic to $S^1\rtimes \mathbb Z/2\mathbb Z$ which is similar to the isomorphism $D_{2n}\approx \mathbb Z/n\mathbb Z\rtimes \mathbb Z/2\mathbb Z$, though. The question is: Which group would you prefer to call $D_\infty$? $S^1\rtimes \mathbb Z/2\mathbb Z$ or $(\mathbb Q/\mathbb Z)\rtimes \mathbb Z/2\mathbb Z$ or maybe $\mathbb Z\rtimes \mathbb Z/2\mathbb Z$? The last at least shares the property of having a cyclic subgroup of index 2; also, it has a corresponding presentation $\langle\,r,s\mid s^2=1, srs=r^{-1}\,\rangle$. That's why (in spite of your idea that is inspired by geometric intuition of polygons converging to the circle) it is the last of these groups, $\mathbb Z\rtimes \mathbb Z/2\mathbb Z$, that is called infinite dihedral group $D_\infty$.