Is a Discrete Group of Rotations Cyclic?

[SNOOTCHIEBOOCHEE]

Homework Statement

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

The Attempt at a Solution

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

But i don't know how to apply this definition to show that G is cyclic. Is this definition even useful?

 

[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?

 

[SNOOTCHIEBOOCHEE]

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

 

[Dick]

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

It sure is.