Quotient of matrix group 
I have checked that $N$ is a normal subgroup of $G$, could any one tell me how to find $G/N$ and what is the known group isomorphic to $G/N$?
Thank you for helping
 A: $G$ itself is isomorphic to ${\mathbb R}_{> 0} \ltimes {\mathbb R}$ (the left ${\mathbb R}_{> 0}$ with multiplication; the right ${\mathbb R}$ with addition). Under this isomorphism, $N$ corresponds to $\{1 \} \times {\mathbb R}$. The quotient, therefore, is isomorphic to ${\mathbb R}_{> 0}$.
More explicitly, consider the subgroup $H$ of $G$ defined by
$$H = \left\{ \begin{pmatrix} a & 0 \\ 0 & a^{-1} \end{pmatrix} \mid a > 0 \right\}.$$
Then every element of $G$ can be written as $hn$ with $h \in H$ and $n \in N$, i.e., $G = HN$; and also $H \cap N = \{ I \}$; and, as the OP already noticed, $N$ is a normal divisor of $G$. Therefore $G = H \ltimes N$. Consequently, $G/N \cong H \cong {\mathbb R}_{> 0}$.
A: Consider the map $\varphi : G \to (\Bbb R_+, \times)$ defined by
$$
\begin{pmatrix} a & b \\ 0 & a^{-1} \end{pmatrix} \mapsto a,
$$
where the range is the multiplicative group of positive real numbers.
It's easy to show that this is a surjective homomorphism. Find the kernel and apply the first isomorphism theorem.
