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Group homomorphism problem

  1. Oct 31, 2007 #1
    1. The problem statement, all variables and given/known data
    Suppose that phi : Z(50)->Z(15) is a group homomorphism with phi(7)=6.
    a) determine phi(x)
    b) Determine the image of phi
    c) determine the kernel of phi
    d) determine (phi^-1)(3))

    2. Relevant equations



    3. The attempt at a solution

    I know how to determine phi: I need to find a multiple of 7 where zero is the remainder. the multiple is 13 . Next I would say 7*13=1 mod 15. => phi(13*7)=phi(1) => 6*13= 78 mod 15 =3. There phi(1)=3. Thus, phi(x)=3*x.

    I'm not sure how to find the image and kernel of phi. I think in order to determine Ker phi, you say Ker phi={x|3x =1}={1/3)}. My result for Ker phi seems incorrect
     
    Last edited: Oct 31, 2007
  2. jcsd
  3. Nov 1, 2007 #2
    Are we dealing with Z50 as a group under addition? Your manipulation of phi seems to suggest so, but if so, the identity would not be 1.

    What is im(phi)? phi(1)=3, so phi(2) = 6, etc. Does this generate a subgroup in Z15? If so, that's the image. This will also tell you the kernel, since the kernel of a homomorphism is also a subgroup.
     
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