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Fixed point free elements

  1. Dec 12, 2011 #1
    1. The problem statement, all variables and given/known data

    What subgroup is generated by the fixed-point-free permutations?



    2. Relevant equations



    3. The attempt at a solution

    I know that the elements that have no fixed points are the ones whose cycle type adds up to n (i.e. all the numbers in {1,...,n} have to be used). I don't know what to do other than just start computing examples and multiplying out every single combination.


    EDIT I think they generate all of Sn. I know this is true up to 5, and whenever we have n odd. (We can generate the transpositions, which in turn generate all of Sn.) For odd n this works by: (1,2)=[(1,2)(3,4,...,n)]^(n-2). I haven't yet found something similar for the case when n is even.
     
    Last edited: Dec 12, 2011
  2. jcsd
  3. Dec 12, 2011 #2

    Dick

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    Science Advisor
    Homework Helper

    Look for other patterns that will let you prove all transpostions are generated. Like for S8, (1,2)(3,4,5)(6,7,8) to the third power. What's a good one for S10? You might have to get a little more creative for S6 but you can certainly do it without multiplying out every single combination.
     
    Last edited: Dec 12, 2011
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