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Math Help - Help Correct My Solution: defining a homomorphism between U(18) and itself

  1. #1
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    Help Correct My Solution: defining a homomorphism between U(18) and itself

    I was asked the following question:

    "Find a homomorphism F:U(18) -> U(18) that has kernel(F)={1,17} and which fulfills F(13)=7."
    Note that U(18)={1,5,7,11,13,17} under multiplication mod(18).

    I was told that my solution, as follows, was incorrect because i had insufficient proof. I was wondering if anyone could help me fill in the gaps or supply a proof of why this method can not work.

    My Solution:

    As the identity of U(18) is 1, the ker(F)={1,17} implies that our mapping must map 1 :-> 1 and 17 :-> 1. We are also given 13 :-> 7. So we have an incomplete mapping that looks like this:

    F: U(18) -> U(18) s.t.
    1 :-> 1
    5 :-> ?
    7 :-> ?
    11 :-> ?
    13 :-> 7
    17 :-> 1

    To figure out where element 5 should map to, note that;
    5= 13*17 mod(18)

    So,
    F(5) = F(13*17)

    If we are to describe this mapping so as to be a homomorphism, we must have;
    F(13*17) = F(13)*F(17)

    However, by the mappings we already have, we know that;
    F(13)*F(17) = 7*1 = 7

    Therefore, we must have;
    F(5) = F(13*17) = F(13)*F(17) = 7*1 = 7.

    By the same logic, note that;
    11 = 5*13 mod(18)
    => F(11) = F(5*13) = F(5)*F(13) = 7*7 = 49 mod(18) = 13

    By the same logic again, note that;
    7 = 11*17 mod(18)
    => F(7) = F(11*17) = F(11)*F(17) = 13*1 = 13

    Therefore, we have a piecewise function F that is a homomorphism by how we defined it, that looks like this;

    F: U(18) -> U(18) s.t.
    1 :-> 1
    5 :-> 7
    7 :-> 13
    11 :-> 13
    13 :-> 7
    17 :-> 1

    This mapping, consequently, is the mapping F: U(18) -> U(18) defined as x :-> x^2.

    Q.E.D.


    WHERE DID I GO WRONG!!!!!!?????!?!????
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  2. #2
    MHF Contributor

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    Quote Originally Posted by DonnMega View Post
    I was asked the following question:

    "Find a homomorphism F:U(18) -> U(18) that has kernel(F)={1,17} and which fulfills F(13)=7."
    Note that U(18)={1,5,7,11,13,17} under multiplication mod(18).

    I was told that my solution, as follows, was incorrect because i had insufficient proof. I was wondering if anyone could help me fill in the gaps or supply a proof of why this method can not work.

    My Solution:

    As the identity of U(18) is 1, the ker(F)={1,17} implies that our mapping must map 1 :-> 1 and 17 :-> 1. We are also given 13 :-> 7. So we have an incomplete mapping that looks like this:

    F: U(18) -> U(18) s.t.
    1 :-> 1
    5 :-> ?
    7 :-> ?
    11 :-> ?
    13 :-> 7
    17 :-> 1

    To figure out where element 5 should map to, note that;
    5= 13*17 mod(18)

    So,
    F(5) = F(13*17)

    If we are to describe this mapping so as to be a homomorphism, we must have;
    F(13*17) = F(13)*F(17)

    However, by the mappings we already have, we know that;
    F(13)*F(17) = 7*1 = 7

    Therefore, we must have;
    F(5) = F(13*17) = F(13)*F(17) = 7*1 = 7.

    By the same logic, note that;
    11 = 5*13 mod(18)
    => F(11) = F(5*13) = F(5)*F(13) = 7*7 = 49 mod(18) = 13

    By the same logic again, note that;
    7 = 11*17 mod(18)
    => F(7) = F(11*17) = F(11)*F(17) = 13*1 = 13

    Therefore, we have a piecewise function F that is a homomorphism by how we defined it, that looks like this;

    F: U(18) -> U(18) s.t.
    1 :-> 1
    5 :-> 7
    7 :-> 13
    11 :-> 13
    13 :-> 7
    17 :-> 1

    This mapping, consequently, is the mapping F: U(18) -> U(18) defined as x :-> x^2.

    Q.E.D.


    WHERE DID I GO WRONG!!!!!!?????!?!????
    why do you think you went wrong somewhere? it looks good to me.
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  3. #3
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    Joined
    Jul 2009
    Posts
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    I was told that, by my assumption that F was to be a homomorphism and how I then built the mapping from there, I was only sufficiently showing that the mapping was a homomorphism for those specific cases. Not every pairing of elements from the domain have been proven to be homomorphic.

    The answer is not wrong, really, it's more of an argument of weather this method can be proven. The only other way I was told this could be done was through a guess and check process. I believe this algorithm works.....
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