Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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


    User Avatar
    Science Advisor
    Homework Helper

    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


    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


    User Avatar
    Science Advisor
    Homework Helper

    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.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook