Why can't I find this group anywhere online?

  • #1
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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
 
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Answers and Replies

  • #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.
 
  • #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.
 
  • #4
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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##?
 
  • #5
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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?
 
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  • #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.
 
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  • #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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