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Isomorphic group needed for cayley table

  1. Mar 25, 2012 #1
    1. The problem statement, all variables and given/known data
    I need to find an isomorphic group for the following group
    F A B C D E F G H < these are the rotations/reflections, f is the operation followed by
    A G D E B C H A F
    B D G F A H C B E
    C E F G H A B C D
    D B A H G F E D C
    E C H A F G D E B
    F H C B E D G F A
    G A B C D E F G H
    H F E D C B A H G
    ^
    These are again the rotations/reflections

    2. Relevant equations



    3. The attempt at a solution
    Someone mentioned that there could be an isomorphism with multiplication under modulo 17 but with my limited knowledge in isomorphic groups I was unable to re-arrange the group as such. Any help would be highly appreciated
     
  2. jcsd
  3. Mar 25, 2012 #2

    Hurkyl

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    (I bet they meant 16, not 17)

    It's a group with 8 elements, right? There aren't very many of those. Have you at least found which one is the identity?
     
  4. Mar 25, 2012 #3
    I haven't found a numerical Identity. I've just been looking at square values for numbers under 17 to see if theres a relationship but I haven't found anything
     
  5. Mar 25, 2012 #4

    Hurkyl

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    Nono, I mean what is the identity element of your group? We can learn a lot by actually working in your group and determining a few simple properties.
     
  6. Mar 25, 2012 #5
    G is the indentity and they all have self inverses
     
  7. Mar 25, 2012 #6

    Hurkyl

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    Right. And since there is only one group with that property, your group has to be isomorphic to it!

    Do you know what group that is? Hint:
    it's abelian


    The squares modulo 17 don't have this property: that group is a cyclic group. The units modulo 16 don't either (mistake on my part: I was thinking of the fact that the units modulo 8 have that property that they're all self-inverses).
     
  8. Mar 25, 2012 #7
    Is there an isomorphic abelian group with 8 elements though. Thats what I need to find. I'm also unaccustomed to cyclic groups, our teacher made us skip it.
     
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