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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


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    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


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    It sure is.
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