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Math Help - Homomorphisms with complex numbers

  1. #1
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    Homomorphisms with complex numbers

    C is the group of all complex numbers under addition.

    a) For each of the following functions, determine whether it is a homomorphism, justifying your answer.

    (i) ∅_1 : C --> C
    z |--> z - 4i


    I have so far come up with two solutions and I am not sure which one is correct. Here they are
    Sol 1:
    ∅_1(z + x) = z + x - 4i
    ∅_1(z) + ∅_1(x) = z - 4i + x - 4i = z + x - 8i
    Since they're not equal, ∅_1 is not a homomorphism

    Sol 2: Let z2 = x + 3i
    ∅_1(z1 + z2) = z + x - 4i + 3i = z + x + 3i
    ∅_1(z1) + ∅_1(z2) = z - 4i + x + 3i = z + x - i
    Since they are equal, ∅_1 is a homomorphism

    Any help will be appreciated.
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  2. #2
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    Quote Originally Posted by storyman01 View Post
    C is the group of all complex numbers under addition.

    a) For each of the following functions, determine whether it is a homomorphism, justifying your answer.

    (i) ∅_1 : C --> C
    z |--> z - 4i


    I have so far come up with two solutions and I am not sure which one is correct. Here they are
    Sol 1:
    ∅_1(z + x) = z + x - 4i
    ∅_1(z) + ∅_1(x) = z - 4i + x - 4i = z + x - 8i
    Since they're not equal, ∅_1 is not a homomorphism

    Sol 2: Let z2 = x + 3i
    ∅_1(z1 + z2) = z + x - 4i + 3i = z + x + 3i
    ∅_1(z1) + ∅_1(z2) = z - 4i + x + 3i = z + x - i
    Since they are equal, ∅_1 is a homomorphism

    Any help will be appreciated.
    How is z+ x+ 3i equal to z+ x- i?
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  3. #3
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    Correction

    ******My apologies Sol 2 should read

    Sol 2: Let z2 = x + 3i
    ∅_1(z1 + z2) = z + x - 4i + 3i = z + x - i
    ∅_1(z1) + ∅_1(z2) = z - 4i + x + 3i = z + x - i
    Since they are equal, ∅_1 is a homomorphism
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