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**storyman01** 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.