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Math Help - Combining Permutations

  1. #1
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    Combining Permutations

    How would you combine the permutation
    (3,4)(2,4)(1,2)
    ?
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  2. #2
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    Re: Combining Permutations

    Trace what happens to each number:
    The first two cycles have no "1" so 1 changes to 2:1->2.
    In the second cycle, 2 changes to 4: 2->4.
    In the first cycle, 3 changes to 4 and then in the second cycle 4 changes to 2, which, in the third cycle, changes to 1: 3->1.
    In the first cycle, 4 changes to 3 which does not then change: 4->3.

    In the basic notation that would be \left(\begin{array}{cccc}1 & 2 & 3 & 4 \\ 2 & 4 & 1 & 3\end{array}\right).
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  3. #3
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    Re: Combining Permutations

    an alternate, but equivalent formulation (and here i am following the opposite convention as HallsofIvy, multiplying right-to-left (like with functional composition)): perform (1 2), which is the function:

    1-->2
    2-->1
    3-->3
    4-->4.

    next, perform the transposition (2 4):

    1-->2-->4
    2-->1-->1
    3-->3-->3
    4-->4-->2.

    finally, we apply (3 4):

    1-->2-->4-->3
    2-->1-->1-->1
    3-->3-->3-->4
    4-->4-->2-->2, which is the permutation (1 3 4 2).

    if, as Halls did, you multiply left-to-right (do (3 4) first, then (2 4), then (1 2), you wind up instead with (1 2 4 3)).

    unfortunately for all of us, permutation multiplication is defined different ways, in different texts.
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