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Groups with new Operations

  1. Jan 24, 2008 #1
    1. The problem statement, all variables and given/known data
    Exercise 1.2:2.
    (i) If G is a group
    Define an operation dG on |G| by dG(x, y) = x*y^-1.
    Does the group given by (G,dG) determine the original group G with *
    (I.e., if G1 and G2 yield the same pair, (G1,dG1) = (G2,dG2) , must G1 = G2?)

    There is a part II, but I would rather focus on I first.

    3. The attempt at a solution

    So, I started by noting that G,dG forces every element to be of order 2 since:
    x dG x = x*x^-1 = e = x dG x^-1 thus x^-1 = x

    Thus G,dG is a klein group. I'm not sure how to proceed, any hint would be appreciated
     
  2. jcsd
  3. Jan 24, 2008 #2

    Dick

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    If you know (G,*) then you know (G,dG). Now ask yourself, if you know (G,dG) then can you figure out what (G,*) is? Hint: dG(x,y^(-1))=x*y.
     
  4. Jan 27, 2008 #3
    Starting from (G,dG)

    dG(x,y) = x*y^-1

    thus

    x*y = dG(x,y^-1)

    Since (G,dG) = (H,dH)

    x*y = dG(x,y^-1) = dH(x,y^-1) = x*y (* in terms of H)

    Thus G = H

    I guess i'm a little confused what it means for G = H. Am I trying to show that they have the same universe and operation or that they are the same upto isomorphism?
     
  5. Jan 27, 2008 #4

    Dick

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    You can only show that they are the same up to isomorphism. That's the strongest sense of 'same' you can ever hope to prove.
     
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