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Math Help - Isomorphism - Group Theory

  1. #1
    Newbie Alborg's Avatar
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    Question Isomorphism - Group Theory

    Are S8, Z8 and Dih(8) isomorphic to each other and why?
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    Quote Originally Posted by Alborg View Post
    Are S8, Z8 and Dih(8) isomorphic to each other and why?
    Since \mathbb{Z}_8 is a communitive group and S_8 (Symmetric group of 8 elements)is not they cannot be isomophic.

    You cannot preserve the group operation.


    Sorry I don't know what Dih(8) stands for

    Good luck.
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    Quote Originally Posted by TheEmptySet View Post
    Since \mathbb{Z}_8 is a communitive group and S_8 (Symmetric group of 8 elements)is not they cannot be isomophic.

    You cannot preserve the group operation.


    Sorry I don't know what Dih(8) stands for

    Good luck.
    Dih(8) is the dihedral group of order 8, i.e. the symmetries of the octagon, with the actions of flipping and rotating 45 degrees.
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    Quote Originally Posted by icemanfan View Post
    Dih(8) is the dihedral group of order 8, i.e. the symmetries of the octagon, with the actions of flipping and rotating 45 degrees.
    Cool I just looked it up in a textbook and it has 16 elements. So it cannot be isomorphic to either of the above two groups because there is not a bijection(1-1 onto function) between the groups.

    I hope this helps.

    Good luck.
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    Quote Originally Posted by TheEmptySet View Post
    Cool I just looked it up in a textbook and it has 16 elements.
    It is conflicting notation. Some books use D_{2n} for the dihedral group because the dihedral group on n verticies is of order 2n.
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