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Math Help - group theory, Orbits of a cube

  1. #1
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    group theory, Orbits of a cube

    I have been trying to work through some group theory questions but have no idea how to solve the ones iv been given. The question is for a tutorial but it is not assessed so I would appreciate any help i can get with it so i can ask my tutor better questions than were do i even start. Hopefully I should be able to solve the other questions on my own with the help i receive on this.

    Question
    Fix a line L through opposite vertices of a cube. Consider the subgroup H of
    the symmetries of the cube generated by g, where g is a rotation by 1/3 of a turn
    about L. Then H acts on the set of vertices of the cube. Describe the orbits.

    Thanks Caitlin
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  2. #2
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    Quote Originally Posted by CHAYNES View Post
    I have been trying to work through some group theory questions but have no idea how to solve the ones iv been given. The question is for a tutorial but it is not assessed so I would appreciate any help i can get with it so i can ask my tutor better questions than were do i even start. Hopefully I should be able to solve the other questions on my own with the help i receive on this.

    Question
    Fix a line L through opposite vertices of a cube. Consider the subgroup H of
    the symmetries of the cube generated by g, where g is a rotation by 1/3 of a turn
    about L. Then H acts on the set of vertices of the cube. Describe the orbits.

    Thanks Caitlin
    It is 120-degree rotations about line L through opposite vertices of a cube. Thus, the order of H is 3. Try first a vertex in a cube and see how it rotates by the action of H. The stabilizer of a vertex that L intersects is H itself.
    Last edited by TheArtofSymmetry; February 23rd 2011 at 06:28 AM.
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