$\Psi:G/N \to (\mathbb R^+,\times)$ with $aN \mapsto x$ is isomorphism

Question

Assume $G=\left\{a+ib\in\mathbb{C} \mid a+ib \ne 0 \right\}$ is a multiplicative group and $N=\left\{a+ib \mid a^2+b^2=1\right\}$,then show that the quotient $G/N$ is isomorphic to the group $(\mathbb R^+,\times)$.

I guess we need to show that there is an isomorphism $\Psi:G/N \to (\mathbb R^+,\times)$ with $aN \mapsto x$

But for example how will we show that the function is homomorphism? As I know if $N \trianglelefteq G$ then for $a_1N,a_2N \in G/N :(a_1N)(a_2N)=a_1a_2N$.

If we show that $N \trianglelefteq G$ then: $$\Psi(a_1N)=\Psi(a_2N)$$

$$\Psi(a_1N)\Psi(a_2N)^{-1}=1$$ $$\Psi(a_1N)\Psi(a_2^{-1}N)=1$$ $$\Psi(a_1a_2^{-1}N)=1$$ However we first need to show thet the function is homomorphism.

But does that help? Or maybe the mapping is not useful?

Answer 1

(First Isomorphism Theorem)
Let $G,H$ be groups and $f:G\rightarrow H$ be a homomorphism. If $f$ is surjective, then $G/{\ker f}$ is isomorphic to $H$.

Hint:

Define $f:G\rightarrow \Bbb{R}^+$ by $$a+ib\mapsto a^2+b^2$$ where $a+ib\in G$.
(i) Show that $f$ is a homomorphism.
(ii) Show that $f$ is surjective.
(iii) Show that $\ker f=N$.
Then by the First Isomorphism Theorem, we have $G/N\cong \Bbb{R}^+$.