2

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!

user1729
  • 31,015
Harry Macpherson
  • 1,824
  • 1
    (1.) You seem to be missing some condition on the group; for instance, the group of all orientation-preserving isometries is certainly not cyclic. Perhaps you want the group to be discrete (i.e., it does not contain arbitrarily small rotations or translations)? (2.) "The group must consist of rotations about a single point" -- I don't see why. For instance, if $t$ is some nontrivial translation map, then how does your proof work for $$, the subgroup generated by $t$? – Srivatsan May 22 '12 at 09:39
  • 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? – Marc van Leeuwen May 22 '12 at 13:50

1 Answers1

2

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.

Phira
  • 20,860