1
$\begingroup$

I have to show that the rotational group of the circle has elements of order n for each n and also elements of infinite order.

A circle has infinite lines of symmetry and so has rotational order of infinity. This is what I know with respect to the question. How should I go further?

$\endgroup$
1
$\begingroup$

Rotation by $\frac{2\pi}n$ has order $n$ and rotation by any angle $\alpha$ that is not a rational multiple of $\pi$ has infinite order.

$\endgroup$
  • $\begingroup$ Where can I read more about this? $\endgroup$ – Mathematicing Jun 2 '15 at 9:36
  • $\begingroup$ But isn't the definition of the order of an element the smallest positive integer such that $$a^n=e$$? $\endgroup$ – Mathematicing Jun 2 '15 at 9:40
  • 2
    $\begingroup$ By convention, Math, if there is no such positive integer, then we say the order is infinite. $\endgroup$ – Gerry Myerson Jun 2 '15 at 10:22
  • $\begingroup$ How would we prove that any $\alpha$ that is not a rational multiple of $\pi$ has infinite order. Wouldnt this rely on the fact that $\pi$ is irrational. Is there a more elementary proof of this statement? $\endgroup$ – Sean Haight Dec 14 '16 at 23:33

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.