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Thread: Cube. Coloring. Isomorphism.

  1. #1
    Junior Member
    Oct 2012

    Cube. Coloring. Isomorphism.

    Let G be a group of rigid motions of cube.

    a) Show that G = S_4

    b) Show that the alternating subgroup A_4 \le S_4 is isomorphic to the group
    of rigid motions of regular

    c)Find cycle index for both S_4 and A_4.

    d) Determine pattern inventory of m coloring of a set
    X = \{d_1,d_2,d_3,d_4\}, where d_i is the i -th diagonal of
    the cube, I(c_1,c_2,...,c_m) (representing colors 1 to m) for
    both S_4 and A_4.

    Please, help!
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  2. #2
    Super Member
    Jan 2008

    Re: Cube. Coloring. Isomorphism.

    What do you need help on? If it's the whole question, I suggest you do some revision BEFORE attempting this question. Otherwise, I can give some hints.

    a) Consider how G acts on - either the set of 6 faces, the set of 12 edges, the set of 8 vertices of a cube. I recommend using the last one. Regardless of your consideration, use the orbit-stabliser theorem to show that |G|=24.

    Now note that vertices which are opposite each other remain opposite of each other under any rigid motion. Thus G acts on the set of pairs of opposite vertices, of which there are 4. This induces a homomorphism G\to S_4, and you just need to show that the map is injective (or surjective) since the orders of G and S_4 agree.

    b) Same idea as before. This time, consider the set of the triangular faces of the tetrahedron (this has cardinality 4). Orbit-stabilizer argument shows that the order of the group is 12. Since G acts on the 4 faces, you have a homomorphism G\to S_4. Now show that the kernel of this map has order 2.

    I suggest you investigate these two parts before moving on to c) and d). I believe you would not need help with the latter two if you understood parts a and b (unless of course, you don't know what cycle indices are. In that case, please revise before attempting this question).
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