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Group Theory

  1. Apr 8, 2009 #1
    Suppose g[tex]\in[/tex] Isom C, z[tex]\in[/tex] C:

    Prove that the g-orbit of z is invariant under g.

    I just need some clarification on what this is asking for:

    1.) Are we assuming that g is a group of the isometries of C under composition?
    2.) To show invariance, would I only have to show that the g-orbit of z [tex]\in[/tex] C?


    Here's my guess at the proof:

    The g-orbit of z is defined as Orbitg(x) = {h(x)|h[tex]\in[/tex] g}, x[tex]\in[/tex] C.
    Now, for all h[tex]\in[/tex] g, h(z)[tex]\in[/tex] C since h[tex]\in[/tex] Isom C, which means h is an isometry from the complex plane to itself. QED.

    Thanks in advance.
     
  2. jcsd
  3. Apr 8, 2009 #2

    Dick

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    What does it mean for h to be an element of g if g is an element of Isom(C)? g isn't a set, is it? Can you clarify your definitions?
     
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