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Basic Group Homomorphism

  1. Jun 7, 2014 #1
    1. The problem statement, all variables and given/known data
    For any integer K, the map ø_k: Z → Z given by ø_k(n) = kn is a homomorpism. Verify this

    2. Relevant equations
    if ø(gh) = ø(g)ø(h) for all g,h in G then the map ø: G → H is a group homomorpism


    3. The attempt at a solution
    So I have barely any linear algebra so many taking this summer course wasn't the best idea, but I have PF so i'm good.
    So...
    if ø(xy) = ø(x)ø(y) for all x,y in Z then the map ø: Z → Z is a group homomorphism. ø_k(x) = kx and ø_k(y) = ky

    but then
    ø(xy) = kxy and ø(x)ø(y) = (xy)k^2

    i'm new to this type of thinking, can anyone help me out here?
     
  2. jcsd
  3. Jun 7, 2014 #2

    pasmith

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    Homework Helper

    The group operation for the integers is addition, not multiplication.
     
  4. Jun 10, 2014 #3
    So you are saying it should read ø(gh) = ø(g) + ø(h)
    ?
     
  5. Jun 10, 2014 #4

    CAF123

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    Gold Member

    No, he meant ##\phi(g + h) = \phi(g) + \phi(h)##. The + in ##\phi(g+h)## is the group operation of the domain (here Z) and the + in ##\phi(g) + \phi(h)## is the group operation of the codomain (here also Z).
     
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