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Permutation help requested

  1. Oct 4, 2007 #1
    1. The problem statement, all variables and given/known data
    let there be a and b (b taking any value) two permutation with the same grade .demonstrate that
    ab=ba <=> a=e(e=the identical permutation) .


    2. Relevant equations
    e=(1234)
    (1234)

    3. The attempt at a solution
    Don't have a clue with what to start/end
     
  2. jcsd
  3. Oct 4, 2007 #2

    EnumaElish

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    How do you define grade?

    Which operation does ab indicate?
     
  4. Oct 4, 2007 #3
    Say you have b= 3 4 2 1 the grade is 4

    ab means a*b that means ab=(a(b(1)), a(b(2)), a(b(3))...... a(b(n)) )
     
    Last edited: Oct 4, 2007
  5. Oct 4, 2007 #4

    EnumaElish

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    Can you give an example with specific a and b, and derive ab?
     
  6. Oct 5, 2007 #5
    a= 1 3 4 2
    b= 2 4 1 3
    ab= 3 2 1 4
     
  7. Oct 5, 2007 #6

    EnumaElish

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    Can you explain why a(b(1)) = 3?
     
  8. Oct 6, 2007 #7
    b(1)=2 , a(2)=3 ===>a(b(1))=3
     
  9. Oct 9, 2007 #8

    EnumaElish

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    I'd start with:

    Let
    a= a1 a2 a3 a4
    and let
    b = b1 b2 b3 b4.

    Then
    ab = a(b(1)) a(b(2)) a(b(3)) a(b(4))
    and
    ba = b(a(1)) b(a(2)) b(a(3)) b(a(4))

    Therefore ab = ba implies
    a(b(1)) = b(a(1)) , ..., a(b(4)) = b(a(4)).

    What are the properties of permutations?
     
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