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Math Help - Structure theorems

  1. #1
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    Structure theorems

    If N \triangleleft G and G/N \cong (\mathbb{Z}, +), show that whenever  H,K \leq G with H \cap K = N , then either H = N or K = N


    Please post a complete proof to this, so I can learn from it. Its a study question for my exam. I really appreciate it.
    Thanks guys
    Last edited by ElieWiesel; December 1st 2009 at 11:08 AM.
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  2. #2
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    Quote Originally Posted by ElieWiesel View Post
    If N \triangleright G and G/N \cong (\mathbb{Z}, +), show that whenever  H,K \leq G with H \cap K = N , then either H = N or K = N


    Please post a complete proof to this, so I can learn from it. Its a study question for my exam. I really appreciate it.
    Thanks guys

    What you're asked to do is equivalent to show that if H\,,\,K\le \mathbb{Z} and H\cap K=\{0\} , then H=\{0\}\,\,\,or\,\,\,K=\{0\}

    Tonio
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  3. #3
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    Hmmmm....I dont really see that Tonio. Can you be more specific....I would appreciate it.

    Thanks
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  4. #4
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    Quote Originally Posted by ElieWiesel View Post
    Hmmmm....I dont really see that Tonio. Can you be more specific....I would appreciate it.

    Thanks

    As G\slash N\cong \mathbb{Z} and N=H\cap K\le H\,,\,K , we get two sbgps H\slash N\,,\,K\slash N \le G\slash N\cong \mathbb{Z}.

    Now, saying that H\cap K=N is the same as saying H\slash N\cap K\slash N=\{\overline{0}\} , with \{\overline{0}\} the unit element of G\slash N\cong \mathbb{Z}...

    Tonio
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