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Dihedral and Symmetric Group

  1. Oct 17, 2009 #1
    Is there a way to prove generally that the Dihedral group and its corresponding Symmetric group of the same order are isormorphic. In class we were only shown a particular example, D3 (or D6 whatever you wish to use) and S3, and a contructed homomorphism, but how could you do it generally? Would you still have to construct a specific map and show that it's a bijective homomorphism? Or can you just simply show there exists at least one isomorphic map between the two?
     
  2. jcsd
  3. Oct 17, 2009 #2

    CompuChip

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    They are not. In general, Dn (or D2n) has 2n elements, whereas Sn has n!.

    Of course, the reason that it works for the triangle group, is that any permutation of its vertices is also a symmetry. However, the permutation group for the vertices of a square is already larger than its symmetry group. For example: if you label the corners 1, 2, 3, 4 in clockwise order, then the symmetry that interchanges 1 with 3 and 2 with 4 does not correspond to any element from D4 (or D8).
     
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