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How can we prove that any finite group of orientation-preserving isometries of the Euclidean plane is cyclic, using the following hint?

'You may assume that given any non-empty finite set E in the Euclidean plane, there is a unique smallest closed disc that contains E.'

I've found a proof not using this (the group must consist of rotations about a single point; pick the one with minimal angle of rotation and use Euclid's algorithm to show that this generates the group). But this took quite a long time, and I wondered if it would be quicker to use the hint.

Many thanks for any help with this!

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    Oriented angles (or the corresponding rotations) form a group isomorphic to $\mathbb R/2\pi\mathbb Z$. How exactly does the Euclidean algorithm operate in terms of angles?2012-05-22

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I will just assume that you forgot to put finite in your first sentence.

Then, your approach is fine (after adding a short comment why translations, and rotations with different centers are excluded).

It's hard to see from your question why the final step "took quite a long time".

You take the rotation with the minimal angle and any other one. Then, either the other one is already a multiple of the minimal one, or do a single step in the Euclidean algorithm (that is, subtracting the smaller angle from the larger one as often as possible) to get a contradiction to the minimality.

So, think that your difficulty is purely technical in the sense that you didn't write down your own argument in the simplest way. I don't even think that this last step would need more space the properly writing out the short comment why you can just regard rotations around a single point.

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    Many thanks - I omitted 'finite' and have added it in. I know my approach is fine, but how is the hint relevant to the first step?2012-05-22