1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Discrete groups of motions

  1. Nov 18, 2008 #1
    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 [tex]\rho_{\theta}[/tex] where [tex]\theta[/tex] 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 [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 dont know how to apply this definition to show that G is cyclic. Is this definition even useful?
  2. jcsd
  3. Nov 18, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper

    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?
  4. Nov 18, 2008 #3
    with that i can show that there is a non zero positive rotation less than theta, a contradiction. Is that it?
  5. Nov 18, 2008 #4


    User Avatar
    Science Advisor
    Homework Helper

    It sure is.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Discrete groups motions Date
[Discrete 2] Permutation/Combination Question Feb 13, 2018
Discrete Pmf of X question Feb 4, 2018
Proving there is a fixed point in a discrete group of rotations Nov 23, 2008
Discrete groups Feb 17, 2008