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Homework Help: Proving there is a fixed point in a discrete group of rotations

  1. Nov 23, 2008 #1
    1. The problem statement, all variables and given/known data

    Let G be a discrete group in which every element is orientation-preserving. Prove that the point group G' is a cyclic group of rotations and that there is a point p in the plane such that the set of group elements which fix p is isomorphic to G'

    3. The attempt at a solution

    Ok since every element of the group is orientation-preserving, we know that there are only translations and rotations. so every element of G can be written as tarhotheta.

    and phi: G----> O

    So the point group G' = phi(G) = rhotheta

    Im guessing i have to use the fixed point theorem for the second part, but i have no clue how to do that.

    Any help is appreciated.
  2. jcsd
  3. Nov 23, 2008 #2
    we need to find a point p such that all g(p)=p for all g in G.

    but i still dont know how to do this.
  4. Nov 24, 2008 #3
  5. Nov 24, 2008 #4


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    What is the definition of "the point group G' "?
  6. Nov 24, 2008 #5
    the image of G in O.

    Oh snap p=(0,0)
    Last edited: Nov 24, 2008
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