# Thread: Homomorphisms with complex numbers

1. ## 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.

2. Originally Posted by 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.
How is z+ x+ 3i equal to z+ x- i?

3. ## 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

### how to identify an homomorphism

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