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Math Help - Homomorphism, kernel, monomorphism :)

  1. #1
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    Wink Homomorphism, kernel, monomorphism :)

    Given f, natural mapping from \mathbb{Z} \text{ to }\mathbb{Z}_n\text{  with }f(m)=r, where r be the remainder if m is divided by n.
    Show that f is homomorphism!
    And determine ker(f).
    Is f monomorphism?
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  2. #2
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by GTK X Hunter View Post
    Given f, natural mapping from \mathbb{Z} \text{ to }\mathbb{Z}_n\text{  with }f(m)=r, where r be the remainder if m is divided by n.
    Show that f is homomorphism!
    And determine ker(f).
    Is f monomorphism?
    rtp f(ab)=f(a)f(b). f(a)f(b)=rs, by definition.

    Now, expand ab=(xn+r)(yn+s) to get the other side of the equality.

    What do you think the kernel of such a mapping should be? Remember, the kernel is everything that is mapped to zero.

    What is f(a) for 1 \leq a \leq n? What does this tell you about surjectivity?
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