# symmetries of a circle and the $O_2$ group: why there are both rotations AND reflections, and not either rotations OR reflections?(geometric view)

I'm reading "Camila Jordan, David Jordan Groups - Modular Mathematics series", section 1.2 Symmetries of a circle.

In this section we consider the unit circle. A typical point on the unit circle has coordinates $$(\cos\theta, \sin\theta)$$ for some $$\theta$$. In contrast to the square, this circle has infinitely many symmetries. They are all the rotations $$rot_{\theta}$$ together with all the reflections $$ref_{\theta}$$. As for the square, these symmetries form a group, the group of symmetries of the circle, but this time the group is infinite. The neutral element is $$rot_0$$. Each rotation $$rot_{\theta}$$ has the rotation $$rot_{-\theta}$$ as inverse while each reflection $$ref_{\theta}$$ is its own inverse. This group is known as orthogonal group $$O_2$$. The group $$O_2$$ acts on the circle by rotations and reflections. Now let us consider the stabilizer of an arbitrary point $$P=(\cos\theta, \sin\theta)$$ on the circle. Which elements of the group $$O_2$$ send $$P$$ to itself? There are just two, namely $$rot_0$$ and $$ref_{2\theta}$$

My question is: why does this $$O_2$$ consist of both rotations and reflections? As if we take any arc on a circle, we can cover all the circle by rotating this arc. Why we would need reflections then?

• It is not a question of needing reflections. The authors are just observing that reflections exist and are symmetries of the circle. And the fact that reflections fix two points of the circle and rotations fix none (except the identity, which fixes everything) shows that reflections are distinct from rotations. Nov 18 '20 at 22:33

• Well, as I said the set of reflections generates the whole group $\mathcal{O}(2)$. However all nontrivial rotations (which are elements of $\mathcal{O}(2)$!) are not reflections. It is an astonishing fact that given any isometry of the circle it is either a reflection or a rotation. Nov 25 '20 at 12:45
• ok, so if the set of reflections generates the whole group $O_2$, even though rotations are not reflections(but any rotation can be acquired as a combination of 2 reflections), why we need rotations in the group itself? Aren't the rotations just aliases for combinations of reflections? Nov 25 '20 at 12:59