$N$ is the Kernel of some homomorphism

Question

How I will show that $N$ is the Kernel of some homomorphism $\phi$ : $G$ $\rightarrow$ $H$. I have no idea.

$G$ := {$f_{a,b}$ : a,b $\in$ $\mathbb{R}$, a $\neq$0 }

$f_{a,b}$ : $\mathbb{R} \rightarrow \mathbb{R}$

$N$:= {$f_{1,b}$ : b $\in$ $\mathbb{R}$}

Thank you

Answer 1

Take $H = \mathbb{R}\setminus \{0\}$ with multiplication and the homomorphism simply takes the $a$ value of $f_{a,b}$.