Question #149664
Let (C*, • ) denote the group of non-zero
complex numbers and
S= {z belongs to C*| IZI = 1 }. Show that
C* / S isomorphic to R+ where (R+, • ) is the group of
positive real numbers.
1
Expert's answer
2020-12-11T12:45:24-0500

Let's consider

ϕ:CR+\phi : \mathbb{C}^* \to \mathbb{R}^+

zzz\mapsto|z|

This function is a surjective homomorphism.

For every xR+x\in\mathbb{R}^+ , we can view this number as an element of C\mathbb{C}^* , xC,x=xx\in\mathbb{C}^*, |x|=x and thus it is surjective.

It is a homomorphism by the properties of complex numbers: ϕ(z1z2)=z1z2=z1z2=ϕ(z1)ϕ(z2)\phi(z_1z_2) = |z_1z_2|=|z_1||z_2|=\phi(z_1)\phi(z_2)

We can also see that Ker(ϕ)={zC:ϕ(z)=z=1}=SKer(\phi) = \{z\in\mathbb{C}^* : \phi(z)=|z|=1 \}=S . Thus by the isomorphism theorem ϕˉ:C/SR+\bar{\phi} : \mathbb{C}^*/S \to \mathbb{R}^+ is an isomorphism and so (C/S,)ϕ(R+,)(\mathbb{C}^*/S,\cdot) \overset{\phi}{\simeq} (\mathbb{R}^+,\cdot) .


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS