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A question about the search for normal subgroups- Conjugacy classes ?

  1. Nov 26, 2009 #1
    A question about the search for normal subgroups- Conjugacy classes!!!?

    1. The problem statement, all variables and given/known data

    Aut(Z*_24) has Conjugacy classes of order 1, 21, 24, 24, 42 and 56.
    Show that Aut(Z*_24) is simple.


    Note: Aut(Z*_24) is sometimes written as U(Z_24)
    Thanks for any idea or answer...


    2. Relevant equations



    3. The attempt at a solution

    I have no idea! At first, why 24 is written 2 times?
     
  2. jcsd
  3. Nov 26, 2009 #2
    Re: A question about the search for normal subgroups- Conjugacy classes!!!?

    It means that it has six conjugacy classes, two of which have the order 24. For starters, what's the definition of a simple group? And how do conjugacy classes relate to that definition?
     
  4. Nov 26, 2009 #3
    Re: A question about the search for normal subgroups- Conjugacy classes!!!?

    Simple group means, if it contains no non-trivial normal subgroups. so, for example, if a group has prime order, then it has no nontrivial normal subgroup.
    and conjugacy classes means for a and b in G, there exist an element g for which,
    g.a.g^(-1)=b
    that means g.a=g.b
    also i know that, Aut(Z*_24) has 168 elements and this is the total of this conjugacy classes. for next step, what should I do?
    Actually i want to know, how can I find conjugacy classes? how did they find this number of conjugacy classes?
     
  5. Nov 26, 2009 #4
    Re: A question about the search for normal subgroups- Conjugacy classes!!!?

    So is there perhaps a connection between the normal subgroups and conjugacy classes? In particular, what would it mean in terms of conjugacy classes, if you had a nontrivial normal subgroup?
     
  6. Nov 28, 2009 #5
    Re: A question about the search for normal subgroups- Conjugacy classes!!!?

    I don't understand what is conjugacy classes and how to compute them.. only i know its definiton.
     
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