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Math Help - Find the order of a permutation

  1. #1
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    Find the order of a permutation

    Ok the question is im given a permutation have to get it as product of disjoint cycles, odd or even, and find the order

    Im not sure how to find the order like when i have
    (1924)(17659)(1238)

    i can get it down to (14)(238765)(9) is the order 9 because it has 9 elements in S9 or does it have something to do with the number of elements in the biggest cycle?
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  2. #2
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    Quote Originally Posted by ChrisBickle View Post
    Ok the question is im given a permutation have to get it as product of disjoint cycles, odd or even, and find the order

    Im not sure how to find the order like when i have
    (1924)(17659)(1238)

    i can get it down to (14)(238765)(9) is the order 9 because it has 9 elements in S9 or does it have something to do with the number of elements in the biggest cycle?
    how you come with this (14)(238765)(9)

    here is my solution

    let us apply them one by one (1924)

    \left(\begin{array}{ccccccccc}1&2&3&4&5&6&7&8&9\\9  &4&3&1&5&6&7&8&2\end{array}\right)

    now (17659)

    \left(\begin{array}{ccccccccc}1&2&3&4&5&6&7&8&9\\9  &4&3&1&5&6&7&8&2\\1&4&3&7&9&5&6&8&2\end{array}\rig  ht)

    now (1238)

    \left(\begin{array}{ccccccccc}1&2&3&4&5&6&7&8&9\\9  &4&3&1&5&6&7&8&2\\1&4&3&7&9&5&6&8&2\\2&4&8&7&9&5&6  &1&3\end{array}\right)

    so (1924)(17659)(1238) = (124765938) has the order 9
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  3. #3
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    Quote Originally Posted by ChrisBickle View Post
    Ok the question is im given a permutation have to get it as product of disjoint cycles, odd or even, and find the order

    Im not sure how to find the order like when i have
    (1924)(17659)(1238)

    i can get it down to (14)(238765)(9) is the order 9 because it has 9 elements in S9 or does it have something to do with the number of elements in the biggest cycle?


    Theorem: when a permutation is written as the product of disjoint cycles (and ANY permutation can be written in this way), it's order is the lowest common multiple of the lengths of those cycles.


    Thus , the order of (14)(238765) is 6 , not 9 ...

    Tonio
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