# Homework Help: Discrete groups of motions

1. Nov 18, 2008

### SNOOTCHIEBOOCHEE

1. The problem statement, all variables and given/known data

Prove that a discrete group G cosisting of rotations about the origin is cyclic and is generated by $$\rho_{\theta}$$ where $$\theta$$ is the smallest angle of rotation in G

3. The attempt at a solution

since G is by definition a discrete group we know that if $$\rho$$ is a rotation in G about some point through a non zero angle $$\theta$$ the the angle $$\theta$$ is at least $$\epsilon$$:|$$\theta$$|$$\geq\epsilon$$

But i dont know how to apply this definition to show that G is cyclic. Is this definition even useful?

2. Nov 18, 2008

### Dick

Ok, if you pick theta to be the smallest angle (which you can do since the rotations are discrete), then all of the rotations n*theta for n an integer are in the group. If that's the whole group, then you are done since it's cyclic. If not there a rotation phi in the group that isn't equal to n*theta for any n. Can you take the next step?

3. Nov 18, 2008

### SNOOTCHIEBOOCHEE

with that i can show that there is a non zero positive rotation less than theta, a contradiction. Is that it?

4. Nov 18, 2008

It sure is.