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Conjugate Elements of a Symmetric Group

  1. Dec 4, 2005 #1
    Is the the following definition correct?

    Two elements a and b of a group G are said to be CONJUGATE if there exists g in G such that [tex]a=gbg^{-1}[/tex].

    For instance, show that all elements in the symmetric group S5 of order 6 conjugate.
     
  2. jcsd
  3. Dec 4, 2005 #2

    matt grime

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    EDIT: Yes, that's the correct definition, as well.


    Again, do it: take an element of order 6, compute its conjugates with a couple of elements and see who to generate all elements of order 6.

    Eg, in S_n n>2, consider (12)(23)(12)=(13), thus it's clear that all elements of order 2 are conjugate
     
    Last edited: Dec 4, 2005
  4. Dec 5, 2005 #3

    matt grime

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    Incidentally, when i say "just do it" i'm not being impatient, it's just that sometimes in maths you sadly just have to get your hands dirty with some calculations.
     
  5. Dec 5, 2005 #4
    I know, I just think I remember a theorem that says two elements in a symmetric group are conjugate if and only if they have the same cycle shape. There are 7 different cycle shapes in S5, I think.
     
  6. Dec 6, 2005 #5

    matt grime

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    Not sure what you're getting at. Perhaps if you realized there was only one cycle shape which corresponded to elements of order 6 that would help.
     
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