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Math Help - Prove that no group of order pq,where p and q are both prime, is simple.

  1. #1
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    Prove that no group of order pq,where p and q are both prime, is simple.

    So this is what I have been asked to prove, and I have included my proof.



    I have now been asked to prove this:



    I feel like the proof for this is exactly the same as the previous proof. Am I missing something? It seems too easy. I think I can use the exact same argument to show that S_q=1 and that therefore the Sylow q-subgroup is normal.
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  2. #2
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    Quote Originally Posted by CropDuster View Post
    So this is what I have been asked to prove, and I have included my proof.



    I have now been asked to prove this:



    I feel like the proof for this is exactly the same as the previous proof. Am I missing something? It seems too easy. I think I can use the exact same argument to show that S_q=1 and that therefore the Sylow q-subgroup is normal.

    No, it's not exactly the same: this time it may be p^2=1+qn\Longrightarrow q\mid (p^2-1) , which
    it's easily possible.

    Tonio
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