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Why can't I find this group anywhere online?

  1. Oct 11, 2015 #1
    http://i.imgur.com/JgpJp03.png

    I've done the Cayley table for the group above and can't find it in any of the group encyclopedias online. I can post it too if you want, but I'll tell you this:

    It is a non abelian group of order 8 with two generators (a,g) such that a^4=Identity and g^4=identity. Plus, it only has three self inverses (apart from the identity).

    No group seems to satisfy this. Anyone knows which group I'm talking about?

    Should I post the Cayley table I've done too?


    edit:

    And here is the Cayley table http://i.imgur.com/VKJr18F.jpg
     
    Last edited: Oct 11, 2015
  2. jcsd
  3. Oct 11, 2015 #2

    Orodruin

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    If you do a little searching, you will find that there are two non-Abelian groups of order 8. The dihedral group ##D_4## and the quaternion group. I will let you figure out which one is the correct one.
     
  4. Oct 11, 2015 #3

    Orodruin

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    Also, your Cayley table does not match the Cayley graph. For example, ##b^2 = d## and not the identity.
     
  5. Oct 11, 2015 #4
    Hmm. I'm checking that ##b^2=d##. For what its worth, I had already checked ##D_4## and the quaternion group and neither match my table. ##D_4## has 6 self inverses, the quaternion group has ##2## (I'm including the identity here).

    Even if ##b^2=d## and my Cayley table is a wrong representation of the diagram, the Cayley table I've shown look quite legit and its making me crazy why I can't find it online.:mad:



    By the way, and this might be a dumb question, but how did you figure out ##b^2=d##?
     
  6. Oct 11, 2015 #5
    I've figured out ##b^2=d## now. Just for the heck of it, as I said, the Cayley table looks legit so why shouldn't it be online? My opinion is that there is something wrong somewhere with my table. After all, it is a proven fact that there is only a certain quantity of groups of order 8, right?
     
    Last edited: Oct 11, 2015
  7. Oct 11, 2015 #6

    Orodruin

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    Your Cayley table does not satisfy associativity. For example, ##(ba)g = 1## while ##b(ag) = d##. Therefore, it does not describe the multiplication table of a group.
     
  8. Oct 11, 2015 #7

    Orodruin

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    And yes, there are only five groups (up to isomorphisms) of order 8. They are the Abelian groups ##C_8##, ##C_4\times C_2##, and ##C_2^3## and the non-Abelian groups already mentioned.
     
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