Artificial intelligent assistant

Why do we need horizontal flip to represent this frieze group? !Frieze group ![enter image description here]( Given this group. Book writes that we need glide reflection and horizontal flip to represent this group. However i think we only need glide reflection. What's the need for horizontal flip in this case ? Also in Cayley diagram why the arrows are in reverse direction in the in the bottom part of the diagram comparison to top part ?

Let's label these..

> $$\cdots\qquad\langle 1\rangle\qquad\qquad \langle 3\rangle\qquad\qquad\langle 5\rangle\qquad \cdots$$
>
> * * *
>
> $$\qquad\cdots\qquad\langle 2\rangle\qquad\qquad\langle 4\rangle \qquad\cdots$$

With glide reflections, we can, say, move $\langle 1\rangle$ to $\langle 2\rangle$, $\langle 2\rangle$ to $\langle 3\rangle$, and so on. _However_ we cannot, say, glide reflect $\langle 1\rangle$ to $\langle 5\rangle$, $\langle 2\rangle$ to $\langle 4\rangle$, etc., while keeping $\langle 3\rangle$ fixed even though this is a symmetry of the image because glide reflections do not have any fixed points.

Does that make sense?

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