Consider the set $G$ of real $2\times2$ matrices of the form $\left(\begin{array}{cc}a&b\\ 0&d\end{array}\right)$, where $a\ne0$, $d\ne0$. Then I have checked that $G$ is a group under matrix multiplication. Consider the set $H=\left\{\left(\begin{array}{cc}1&b\\ 0&1\end{array}\right)~:~b\in\Bbb{R}\right\}$. I have also checked $H\trianglelefteq G$.
Now how do I show $G/H$ is abelian.
I have tried showing the the map $g\mapsto g^{-1}$ is an isomorphism on this quotient but to no avail.
Hint: define $\phi:G\to \mathbb{R}^{\times}\times\mathbb{R}^{\times}$ by $$ \phi\Big(\begin{bmatrix}a&b\\0&d\end{bmatrix}\Big)=(a,d)$$ and show that $\phi$ is a surjective group homomorphism with kernel $H$.