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Math Help - prove homomorphism and determine Kernel

  1. #1
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    Unhappy prove homomorphism and determine Kernel

    Phi is the mapping from Z to Z_12 by for all a that is a member of Z, phi(a)=[3a]mod 12.
    -Prove that phi is homomorphism
    -Determine Ker(phi)

    This is what I have, but don't think it is right, can someone help me Please?

    Homomorphism:
    phi(ab)=(ab)^12=a^12 * b^12=phi(a)phi(b)

    And the Ker(phi)=<cos 120 + i sin120>

    I am completely lost and dont know if I did it right!?
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  2. #2
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    Quote Originally Posted by mandy123 View Post
    Homomorphism:
    phi(ab)=(ab)^12=a^12 * b^12=phi(a)phi(b)
    You have \phi (ab) = [3ab]_{12} and \phi(a) \phi(b) = [3a]_{12} \cdot [3b]_{12}.
    To be a homomorphism you need [3a]_{12}\cdot [3b]_{12} = [3ab]_{12}.
    This seems to be work, therefore it cannot be a homomorphism.
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  3. #3
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    What about under addition would it be homomorphism then?

    What about the Ker(phi) under the operation addition?
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  4. #4
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    Quote Originally Posted by mandy123 View Post
    What about the Ker(phi) under the operation addition?
    By definition \ker (\phi) = \{ x\in \mathbb{Z} : [3x]_{12} = [0]_{12} \}.

    Therefore, 3x\equiv 0 (\bmod 12) \implies x\equiv 0 (\bmod 4).
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  5. #5
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    Thank you so much, I am starting to understand this stuff!!!!
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