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Math Help - Quick verification of something...

  1. #1
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    Quick verification of something...

    Is this true?
    H, N, G are groups

    H \triangleleft N \triangleleft G implies H \triangleleft G
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  2. #2
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    Quote Originally Posted by ElieWiesel View Post
    Is this true?
    H, N, G are groups

    H \triangleleft N \triangleleft G implies H \triangleleft G

    It is false: \{(1),(12)(34)\}\,\triangleleft\, \{(1),(12)(34),(13)(24),(14)(23)\}\,\triangleleft\  , A_4\,,\,\,but\,\,\,\{(1),(12)(34)\}\,\ntrianglelef  t\, A_4

    Tonio
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  3. #3
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    What if N is cyclic? Does it then imply that H is abelian, and hence H a subgroup of G is normal in G?
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    Quote Originally Posted by ElieWiesel View Post
    What if N is cyclic? Does it then imply that H is abelian, and hence H a subgroup of G is normal in G?

    Yes, then H is normal in G, but not your "hence", because if N, H are abelian then still it isn't true that H must be normal, as my counterexample shows. I leave it to you to prove that if N is cyclic then H is normal in G.

    Tonio
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