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Isomorphic automorphisms

  1. Jun 11, 2007 #1
    1. The problem statement, all variables and given/known data
    Give an example of an abelian and a non-abelian group with isomorphic automorphism groups.

    3. The attempt at a solution


    My classmate talked to our professor and he hints that Z2 xZ3 and D3 (or D6..depends on your notational preference it's the triangle) MIGHT be correct...prove or disprove.....

    I see that Z2xZ3 is of order 6 and so is D3. So lovely, off to a good start. At least we start with the same group order.

    I need to find the Automorphism group of each set to show that these 2 are isomorphic...(IF they even are) and this is where I can't go any further.

    D3 is not abelian but Z2 x Z3 IS abelian and I'm looking at the automorphism groups of each one.

    How do I get these automorphism groups? I just am drawing a blank here. We know that each one has the identity Aut, but then how do we define the other ones. We've confused ourselves!

    EDIT: So are the automorphisms of D3 [tex]1, r, r^ 2, a, ra, r^ 2a[/tex] where 1 is the identity and r is a rotation by 120 degrees and a is a flip through the vertex angle? Or is it something else?

    CC
     
    Last edited: Jun 11, 2007
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  3. Jun 11, 2007 #2

    Hurkyl

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    The problem is small enough to be solved with sheer brute force.

    Each of these groups has 6 elements, and can be presented with two generators. Therefore, there are only thirty-six ways to map the two generators back into the group; you can go through each one and see if it extends to a homomorphism, and if that homomorphism is an automorphism.

    After doing the first few, hopefully you'll get some ideas on how to greatly accelerate the search... :smile:
     
  4. Jun 12, 2007 #3

    matt grime

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    Those are the elements of D_2.3, not automorphisms of it. An automorphism of a group is an isomorphism from it to itself.

    Auts of Z_2 x Z_3=Z_6 are easy, since that is a cyclic group generated by g say of order 6. Any hom of Z_6 is determined by where it sends g, and there are 6 possiblities - how many of those give isomorphisms?
     
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