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Math Help - Order of Permutation

  1. #1
    NYMFan
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    Order of Permutation

    Hello. Could someone help me with this?

    Find the order of the permutation, sigma =
    (1 2 3 4 5 6 7 8
    3 1 5 6 2 7 8 4)
    and write it as a product of cycles.

    It's for a graduate school project and honestly I have never seen a problem like this before. So I appreciate any guidance.
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  2. #2
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    Quote Originally Posted by NYMFan View Post
    Hello. Could someone help me with this?

    Find the order of the permutation, sigma =
    (1 2 3 4 5 6 7 8
    3 1 5 6 2 7 8 4)
    and write it as a product of cycles.

    It's for a graduate school project and honestly I have never seen a problem like this before. So I appreciate any guidance.
    So you have the premutation on \{1,2,3,4,5,6,7,8\} defined as,
    \sigma = \left( \begin{array}{cccccccc}1&2&3&4&5&6&7&8\\ 3&1&5&6&2&7&8&4 \end{array}\right)
    This is an element of a finite group S_8 thus it has an order. We need to find the smallest positive integer n such that \sigma^n is the identity premutation.
    Observe the following,
    3\to 5\to 2\to 1
    1\to 3\to 5\to 2
    5\to 2\to 1\to 3
    6\to 7\to 8\to 4
    2\to 1\to 3\to 5
    7\to 8\to 4\to 6
    8\to 4\to 6\to 7
    4\to 6\to 7\to 8
    This show that n=3 is the order.

    To express it as a product of disjoint cycles we find the orbits of \sigma which are,
    \{1,3,5,2\}
    \{4,6,7,8\}
    Thus,
    \sigma=(1,3,5,2)(4,6,7,8)
    (Note although the binary operation on the premutation group is not commutative here it is and I could have expressed the product the other way aroung because the cycles are disjoint).
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