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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?

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    $\begingroup$ 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. $\endgroup$
    – Derek Holt
    Nov 18 '20 at 22:33
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Think of the circle as being oriented, say counterclockwise. Then rotations preserve this orientation, while reflections change it. Hence there are at least two kind of symmetries...

Note however that rotations can be obtained by composing two reflections about different axes. So in some sense reflections alone do suffice.

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  • $\begingroup$ Reflection can't be acquired using rotations. But rotation can be acquired using reflections, indeed. Then why we need rotations? Why this group can't be constructed using reflections only? $\endgroup$ Nov 25 '20 at 12:37
  • $\begingroup$ 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. $\endgroup$
    – PrudiiArca
    Nov 25 '20 at 12:45
  • $\begingroup$ 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? $\endgroup$ Nov 25 '20 at 12:59
  • $\begingroup$ You dont need them to do the things geometric things you want, but they exist and are very convenient to have. $\endgroup$
    – PrudiiArca
    Nov 25 '20 at 13:11
  • $\begingroup$ Another side of the same problem: Every natural number is some combination of 1s. But do you really want to say the other numbers do not matter? $\endgroup$
    – PrudiiArca
    Nov 25 '20 at 13:20

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