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Orbits of Transitive Group Actions

  1. Apr 7, 2009 #1
    1. The problem statement, all variables and given/known data
    Suppose G is a finite group, which acts transitively on a set X and let H be a normal subgroup of G. Show that the size of the orbits of the action of H on X are of the same size.


    2. Relevant equations



    3. The attempt at a solution
    I haven't been able to get very far with this. The only fact I've gotten is that |H(x)| divides |G|, but I am not sure that is even relevant in this case.
     
  2. jcsd
  3. Apr 8, 2009 #2
    First think about this example.

    Let G=group of all rotations in the plane about the origin.

    Let H=some subgroup, e.g. [tex]H=\{ R_0, R_{120}, R_{240} \}[/tex].

    Let X=the unit circle with center at origin.

    Let [tex]x=x_0=[/tex] some fixed point [tex] p_0[/tex] on the unit circle, e.g. [tex] p_0=(1,0)[/tex].

    Let [tex]y=[/tex] some other point on the unit circle.

    Draw a picture of Hx.

    Now, what if you want to find Hy, but for some crazy reason you don't know how to find it directly. You only know how to apply the rotations of H to the point x, not any other point. Geometrically, describe how to find Hy. Use the word "rotate."
     
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