Thread: group homomorphism examples with comlpex numbers

1. group homomorphism examples with comlpex numbers

I think i get how to prove a homomorphism just when they add in complex numbers i panic!
I need to prove whether the following are a homomorphism

a) sin(Rez)
Complex numbers under multiplication
b) e^z
c) z/z
Any help would be very gratefully appreciated.
Many thanks

2. Re: group homomorphism examples with comlpex numbers

Well, do you know the definition of "homomorphism"? You say "I think i get how to prove a homomorphism" so I assume you do. You prove that a function from one group to another by showing that each point in the definition is true. For a function, f, from one group, A, to another, B, you must prove
For any x, y in A, f(x* y)= f(x)* f(y) where "*" is the group operation.
It is also true that $\displaystyle f(0_A)= 0_B$ or $\displaystyle f(1_A)= 1_B$ where $\displaystyle 0_A, 0_B, 1_A$ and [tex]1_B[/tex[ are the additive and multiplicative identities of A and B and that [tex]f(x^{-1})= (f(x))^{-1} but, if I remember correctly those can be derived from the first two requirements so only those two need be shown.

But your exercises are not complete- you give only a single group. In the second problem you say
"Complex numbers under multiplication
b) e^z"

If you mean from the set of complex numbers, with multiplication as the operation to the set of complex numbers, with multiplication as the operation, this is not a homomorphism. But if you mean from the set of complex numbers, with multiplication as the operation to the set of complex numbers, with addition as the operation, then it is. That is because $\displaystyle (e^{z_1})(e^{z_2})= e^{z_1+ z_2}$.

3. Re: group homomorphism examples with comlpex numbers

Should i be using: Sin (Rez)=Sin(x)
Then: Sin (x1+x2) Which wouldn't be a homomorphism according to angle sum and difference identities. Or am i completely on the wrong trail of thought.
Many thanks

4. Re: group homomorphism examples with comlpex numbers

That is correct. The first is not a homomorphism. But you have not addressed the more important point- these functions are from what group to what group?

5. Re: group homomorphism examples with comlpex numbers

Originally Posted by HallsofIvy
you have not addressed the more important point- these functions are from what group to what group?
I think that the attachment in the OP does just that.
It says $\phi_1:\mathbb{C}\to\mathbb{C}$; $\phi_2:\mathbb{C}^*\to\mathbb{C}^*$ and; $\phi_3:\mathbb{C}^*\to\mathbb{C}^*$
But it does not say what $\mathbb{C}^*$ is (multiplication ?)

6. Re: group homomorphism examples with comlpex numbers

In this question, C∗ is the group of non-zero complex numbers under multiplication,
and C is the group of all complex numbers under addition.

7. Re: group homomorphism examples with comlpex numbers

Originally Posted by smclaughlin78
In this question, C∗ is the group of non-zero complex numbers under multiplication,
and C is the group of all complex numbers under addition.
$\phi_2:\mathbb{C}\to\mathbb{C}$ defined by $z\mapsto \sin(\Re(z))$
You seem confused about the fact that $\sin(\Re(z))= \sin(x)$ As you know $\sin(z)$ is not additive.

$\phi_3:\mathbb{C}^*\to\mathbb{C}^*$ defined by $z\mapsto \exp(z)$
$\exp(z)\cdot\exp(w)\ne\exp(z\cdot w)$

$\phi_4:\mathbb{C}^*\to\mathbb{C}^*$ defined by $z\mapsto \dfrac{z}{\overline{z}}$ Can you show that $\overline{z\cdot w}=\overline{z}\cdot\overline{w}~?$