# Rotational group of a circle

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?

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.
• But isn't the definition of the order of an element the smallest positive integer such that $$a^n=e$$? – Mathematicing Jun 2 '15 at 9:40
• 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? – Sean Haight Dec 14 '16 at 23:33