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Math Help - Counting formula to represent the decomposition into orbits.

  1. #1
    Senior Member Pinkk's Avatar
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    Counting formula to represent the decomposition into orbits.

    Let G be the group of rotational symmetries of a cube, let G_{v}, G_{e}, G_{f} be the stabilizers of a vertex v, an edge e, and a face f of the cube, and let V, E, F be the sets of vertices, edges, and faces, respectively. Determine the formulas that represent the decomposition of each of the three sets into orbits for each of the subgroups.

    I know that the order of the orbit of an element of a set on which a group G operates is equal to the index of the stabilizer, but other than that I'm having a hard time figuring out how to go about this or what is really being asked here. Any help would be appreciated.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Pinkk View Post
    Let G be the group of rotational symmetries of a cube, let G_{v}, G_{e}, G_{f} be the stabilizers of a vertex v, an edge e, and a face f of the cube, and let V, E, F be the sets of vertices, edges, and faces, respectively. Determine the formulas that represent the decomposition of each of the three sets into orbits for each of the subgroups.

    I know that the order of the orbit of an element of a set on which a group G operates is equal to the index of the stabilizer, but other than that I'm having a hard time figuring out how to go about this or what is really being asked here. Any help would be appreciated.
    This is a strange question. So, I assume what it's saying is that the group G acts on the set V say by taking a vertex v\in V and some g\in G and defining g v to be the vertex that v ends up at after taking the cube performing the rotation associated to g to it. You clearly haven that V is broken up into orbits associated to this action.


    This question is really not worded well in my opinion since I really have no idea what they want.
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  3. #3
    Senior Member Pinkk's Avatar
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    Me neither, but I'm still not sure how the orbits look for the vertices. For any vertex of the cube, isn't there a rotation that brings it to any other rotation, and so the orbit is the whole set of vertices?
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Pinkk View Post
    Me neither, but I'm still not sure how the orbits look for the vertices. For any vertex of the cube, isn't there a rotation that brings it to any other rotation, and so the orbit is the whole set of vertices?
    Precisely, that is what's strange--the group acts transitively on the set of vertices.
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  5. #5
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    ...so if G acts transitively on the vertices, the orbit of one (and thus any) vertex v has order 8. which means that the index of the stabilizer [G:Gv] = 8, so there are 8 cosets of Gv in G.

    if one knows that the rotational symmetry group of the cube is S4, this tells you that Gv has order 3.

    G also acts transitively on the faces, so Gf (for any face f) has order 4.

    finally, G also acts transitively on the edges, so Ge (for any edge e) has order 2.

    (the class equation for these subsets of {faces, vertices, edges} is particularly simple).

    it occurs to me, that it is possible you are being asked to compute the orbits of V,F,E under the respective actions induced by each group:

    Gv,Gf, and Ge, for some particular stabilizer in each set. this is somewhat of a different matter.

    for example, say V = {v1,v2,v3,v4,v5,v6,v7,v8}. Gv1 fixes v1, so it's orbit is: {v1}. if the opposite vertex is v7, Gv1 also fixes v7,

    so its orbit is: {v7}. the other two orbits have to have order 3, since there are no other points fixed by Gv1, except v1 and v7,

    and the size of the orbits has to divide |Gv1| (it helps to think about WHICH rotations Gv1 must be: rotations about the axis between v1 and v7).

    you can carry out a similar analysis for E and F.
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  6. #6
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Deveno View Post
    ...so if G acts transitively on the vertices, the orbit of one (and thus any) vertex v has order 8. which means that the index of the stabilizer [G:Gv] = 8, so there are 8 cosets of Gv in G.

    if one knows that the rotational symmetry group of the cube is S4, this tells you that Gv has order 3.

    G also acts transitively on the faces, so Gf (for any face f) has order 4.

    finally, G also acts transitively on the edges, so Ge (for any edge e) has order 2.

    (the class equation for these subsets of {faces, vertices, edges} is particularly simple).

    it occurs to me, that it is possible you are being asked to compute the orbits of V,F,E under the respective actions induced by each group:

    Gv,Gf, and Ge, for some particular stabilizer in each set. this is somewhat of a different matter.

    for example, say V = {v1,v2,v3,v4,v5,v6,v7,v8}. Gv1 fixes v1, so it's orbit is: {v1}. if the opposite vertex is v7, Gv1 also fixes v7,

    so its orbit is: {v7}. the other two orbits have to have order 3, since there are no other points fixed by Gv1, except v1 and v7,

    and the size of the orbits has to divide |Gv1| (it helps to think about WHICH rotations Gv1 must be: rotations about the axis between v1 and v7).

    you can carry out a similar analysis for E and F.
    Gee whiz, is that really what they wanted you to do. How sadistic. And what's wrong with you! How do you have enough time to write out all these responses! Don't you have math of your own you should be doing? haha
    Last edited by mr fantastic; July 10th 2011 at 04:22 PM.
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  7. #7
    Senior Member Pinkk's Avatar
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    I sorta follow what you're saying Deveno, but I don't see how that is "determining formulas." But that could just be the crappy wording of the problem, because now looking at it, aren't the formulas just |O_{v}| = [G:V], and so on for the edges and faces? I am having a really hard time grasping the concepts of orbits, stabilizers, finite subgroups of rotation groups, etc.
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