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Homework Help: Show that S_N is of order N!

  1. Sep 15, 2009 #1
    1. The problem statement, all variables and given/known data

    Show that the symmetric group (permutation group) S_N is of order N!

    2. Relevant equations

    3. The attempt at a solution

    I can't get started on how to prove this. I understand that if

    n=2 S_2= {E, (12)}
    n=3 S_3={E, (12), (13), (23), (123), (132)}

    and so on. This makes this seem kind of intuitive. However I can't even get started on proving it for N. Could somebody please help me get started?
  2. jcsd
  3. Sep 15, 2009 #2
    Given {1, 2, ..., n-1, n}, you want to construct a bijection onto itself. So take the first element, 1. There are n choices of elements it can map to. Then consider 2. There are n-1 elements it can map to (since the function is 1-1 and one of the n elements is already 'taken' by 1)
  4. Sep 15, 2009 #3
    Got it. Thanks.
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