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Math Help - Automorphisms of the Dihedral Group D8

  1. #1
    Super Member Bernhard's Avatar
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    Automorphisms of the Dihedral Group D8

    Dummit and Foote Section 4.4 Automorphisms, Exercise 3 reads as follows:

    ==================================================

    Prove that under any automomorpism of D_8, r has at most two possible images and s has at most 4 possible images. Deduce that |Aut( D_8)| \leq8

    ================================================== =

    Can anyone help with the above problem? Be grateful for help!



    The notation of D&F regarding the construction of D_8 is as follows:

    Fix a square centred at the origin in an x,y plane and label the vertices from 1 to 4 in a clockweise manner. Let r be the rotation clockwise about the origin through \pi /2 radian. Let s be the reflection about the line of symmetry through vertex 1 and the origin.

    Then

    D_8 = {1, r, r^2, r^3, s, sr, s r^2, s r^3}


    Peter
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  2. #2
    MHF Contributor Amer's Avatar
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    Re: Automorphisms of the Dihedral Group D8

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    Re: Automorphisms of the Dihedral Group D8

    to fill in the gaps of the link Amer gave (which is a great site to look up various math proofs), note that Z(D_8) = \{1,r^2\}.

    if you really feel ambitious, try to prove that \mathrm{Aut}(D_8) \cong D_8.
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  4. #4
    Super Member Bernhard's Avatar
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    Re: Automorphisms of the Dihedral Group D8

    Thanks for the link - looks really helpful!!!

    Peter
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  5. #5
    MHF Contributor Amer's Avatar
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    Re: Automorphisms of the Dihedral Group D8

    Quote Originally Posted by Bernhard View Post
    Thanks for the link - looks really helpful!!!

    Peter
    yea it is very it contains alot of problems from Dummit Footie Abstract Algebra
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