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Homework Help: Sylow theory

  1. Mar 4, 2008 #1
    [SOLVED] Sylow theory

    1. The problem statement, all variables and given/known data
    Find all Sylow 3-subgroups of S_4 and demonstrate that they are all conjugate.

    2. Relevant equations

    3. The attempt at a solution
    I found all the Sylow 3-subgroups, but I am having trouble finding the element that conjugates them. For example, how do you find the element conjugates that <(1,2,3)> and <(2,3,4)>? I could just try all the elements, but there are 24, so that is probably a bad idea...
  2. jcsd
  3. Mar 4, 2008 #2
    well, you can just try all elements, leave a few by symmetry.

    But you'd only have to do them all once, because if you did it correct you'll see all the sylow subgroups emerge =)

    If choose carefully you'll probably only have to do a bit more than there are subgroups in the class.
    Last edited: Mar 4, 2008
  4. Mar 4, 2008 #3


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    Homework Helper

    You could also try to solve the general case:
    [tex]p (1, 2, 3) p^{-1} = (a,b,c)[/tex]
    [tex]a\neq b \neq c[/itex]
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