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Conjugate Subgroups

  1. Sep 10, 2011 #1
    1. The problem statement, all variables and given/known data
    How would I go about proving that if K = gHg-1, for some g [itex]\in[/itex]G, where K and H are both subgroups of G with a prime number of elements, then K = H?


    2. Relevant equations
    I've tried to prove it by saying that if K = gHg-1 then Kg = gH, and since H = gHg-1, then Hg = gH also, so Hg = Kg, and hence H = K. I don't think that this proof is valid, unfortunately. And I've just realised that H does not necessarily equal gHg-1 unless g is in the normaliser of H. :(

    3. The attempt at a solution
    This is actually something which I am attempting to prove in order to prove something else, so I'm not even sure if what I'm trying to prove holds at all. I just need K = H for my proof to work.
     
  2. jcsd
  3. Sep 10, 2011 #2

    Dick

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    Take G=S3, the symmetric group on three elements. Take K={e,(12)} and H={e,(13)}. The subgroups both have order 2 (a prime) and they are conjugate via g=(23), but they aren't equal.
     
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