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Math Help - Group action

  1. #1
    Junior Member maths900's Avatar
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    Group action

    How would you show the group action where G is the symmetries of a regular octagon and X is a set of all possible coulourings of a regular octagon where each edge is coloured one of two colours?
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  2. #2
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    Quote Originally Posted by maths900 View Post
    How would you show the group action where G is the symmetries of a regular octagon and X is a set of all possible coulourings of a regular octagon where each edge is coloured one of two colours?
    Hi, I have tried some of key steps of your problem. You need to fill the remaining steps. Assume the same color can be used on any number of edges.

    Burnside's lemma

    r \cdot |G| = \sum_{g \in G}|X_g|.

    The number of distinguishable regular octagons correspond to the number of orbits under G in a G-set X.

    To apply symmetry to a regular octagon, we use G = D_8=\{1, r, r^2, r^3, ..., r^7, s, sr, sr^2, sr^3, ..., sr^7\}.

    Now we need to compute |X_g| for each sixteen elements g in D_8.

    |X_1| = 2^8. (Every regular octagon is left fixed by the identity element)
    |X_r| = |X_{r^7}| = 2. (To be invariant under r, r^7, all edges should be the same color. There are two colors, so two choices)
    |X_{r^4}| = 6. (To be invariant under r^4, all edges should be same colors (oooooooo, xxxxxxxx), or different adjacent colors (oxoxoxox, xoxoxoxo), and (ooxxooxx, xxooxxoo). Here , (oxoxoxox) denotes edge 1 (between vertex 1 and 2) is colored o, edge 2 is colored x (between vertex 2 and 3), and so on.

    |X_s| = 4. Diagonal flip : (Possible choices: oooooooo, xxxxxxxx, oxoxoxox, xoxoxoxo}
    .....

    After you find all |X_g| values, you need to sum up those |X_g| values
    \sum_{g \in G}|X_g| = \{2^8 + 2 + .. + 6 + ..+ 4 + ...\}

    Finally, the number of orbits N (distinguishable regular octagons) can be acquired.

     N = \frac{\sum_{g \in G}|X_g|}{|D_{8}| = 16}.
    Last edited by aliceinwonderland; October 15th 2009 at 02:01 PM. Reason: Seems a bit complicated to count all possible choices (added and removed some choices).
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