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Homework Help: Application of Burnside's Theorem, Colorings

  1. Apr 28, 2008 #1
    1. The problem statement, all variables and given/known data
    I have multiple problems, all dealing with Burnside's Theorem, perhaps help on one would help explain the others. How many ways may the faces of a cube be colored using 3 colors, up to symmetry of the cube. And, how many ways may the faces of a dodecahedron be colored using 5 colors up to symmetry.

    2. Relevant equations

    Burnside's Theorem: # colorings= 1/(group of symmetry)*Sum of number of fixed colorings

    3. The attempt at a solution

    The symmetry groups are S4 and A5 respectively. The part that I get hung up on is the number of fixed colorings. I know for the identity permutation on the cube, all colorings are fixed, but I don't know how the other rotations work. Please help!
  2. jcsd
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