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Math Help - symmetric groups 2

  1. #1
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    symmetric groups 2

    Evaluate all the pwers of each permutation s (ie find s^k for all k).

    a) 1 2 3 4 5 6
    2 3 4 5 6 1

    b) 1 2 3 4 5 6 7
    2 1 3 4 6 5 7

    c) 1 2 3 4 5 6
    6 4 5 2 1 3
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  2. #2
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    Quote Originally Posted by mpryal View Post
    Evaluate all the pwers of each permutation s (ie find s^k for all k).

    a) 1 2 3 4 5 6
    2 3 4 5 6 1
    I do the first one partially and you try the other ones.
    Let s the permutation \begin{bmatrix} 1&2&3&4&5&6\\2&3&4&5&6&1\end{bmatrix}.

    We have:
    s^2(1) = s(s(1)) = s(2) = 3
    s^2(2) = s(s(2)) = s(3) = 4
    s^2(3) = s(s(3)) = s(4) = 5
    s^2(4) = s(s(4)) = s(5) = 6
    s^2(5) = s(s(5)) = s(6) = 1
    s^2(6) = s(s(6)) = s(1) = 2

    Thus, s^2 is the permutation \begin{bmatrix} 1&2&3&4&5&6\\ 3&4&5&6&1&2 \end{bmatrix}

    We have:
    s^3(1) = s(s^2(1)) = s(3) = 4
    s^3(2) = s(s^2(2)) = s(4) = 5
    s^4(3) ...
    ...

    Thus, s^3 is the permutation \begin{bmatrix} 1&2&3&4&5&6\\ 4&5&..&..&..&.. \end{bmatrix}

    Fill in the ".." 's.
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