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Permutations and Transpositions

  1. Dec 7, 2003 #1
    Hello,

    I am a little confused about an example. By definition,

    A cycle of m symbols CAN be written as a product of m - 1 transpositions.

    (x1 x2 x3 ... xn) = (x1 x2)(x1 x3)...(x1 xn)


    Now

    Express the permutation (23) on S = {1,2,3,4,5} as a product of transpositions.


    (23) = (12) o (23) o (13) = (12) o (13) o (12)


    I can see how it works. But based on the def. I don't see how they came up with the answer. I know this is simple but I don't see it. What the hey?
     
  2. jcsd
  3. Dec 7, 2003 #2

    Hurkyl

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    I'm a little confused; (2, 3) is a product of transpositions...

    can you provide a little more of the context?
     
  4. Dec 7, 2003 #3
    I am confused too.

    This is in Schaum's Outlines of Modern Abstract Algebra. It is in Chapter 2: Relations and Operations, under the section Permutations.

    The question/ example above is exactly as it is in the book.
    I know that a permutation can be expressed as a product of transpositions. And that there can be more than one way to express a permutation as a product of transpositions. I think that is what they are trying to show.

    However I don't understand the method in which they selected these particular transpositions to express the permutation (23). I can see that it works out. But why/how did they know that (23) was a product of the above transpositions? Trial and error?
     
  5. Dec 7, 2003 #4

    Hurkyl

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    Ah, an example; it makes more sense now.

    Anyways, I can see an algorithmic procedure that gives you the second example, but I'm tongue-tied trying to explain it... if you limit yourself to the condition that each transposition must have '1' in it, you could probably figure the procedure out for yourself.


    I can motivate the first one from products of transpositions:
    (12) (23) = (23) (13)
    so
    (23) = (12) (23) (13)

    then again, they might simply just be examples without expecting any motivation.
     
  6. Dec 7, 2003 #5
    It just confused me since the way they got the product of transpositions for (23) wasn't based on the defintion.

    (x1 x2 x3 ... xn) = (x1 x2)(x1 x3)...(x1 xn)

    I mean, using the def. I couldn't see how one could come up with

    (23) = (12) o (23) o (13) = (12) o (13) o (12).

    Thanks Hurkyl.
     
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