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Math Help - Isometries

  1. #1
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    Isometries

    Show that D_4 (isometries of a square with vertices 1,2,3,4) is not normal in S_4

    (Permutation of {1,2,3,4})
     |D_4|=8
     |S_4|=4!

    Any help is appreciated! Thanks!
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  2. #2
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    Quote Originally Posted by Linnus View Post
    Show that D_4 (isometries of a square with vertices 1,2,3,4) is not normal in S_4

    (Permutation of {1,2,3,4})
     |D_4|=8
     |S_4|=4!

    Any help is appreciated! Thanks!
    Identify D_4 with a definite subgroup of S_4 , since you want to show it is not normal there: it will contain two 4-cycles. Now conjugate one of these 4-cycles by a nicely chosen transposition and get a new 4-cycle that isn't contained in your D_4 ...

    Another way: D_4 is a Sylow 2-subgroup of S_4 so it is normal there iff there's one unique subgroup of its order (8); Well, show that there are more than one...

    Tonio
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