I am working on the below problem:
Let $G = \{\begin{pmatrix} a & c\\ 0 & b \end{pmatrix}\ | a,b,c \in \mathbb{R}, ab \neq 0\}$, a group under matrix multiplication. Find a normal subgroup $N$ of $G$ such that $G/N$ is isomorphic to $\mathbb{R}^{\times} \times \mathbb{R}^{\times}$, where $\mathbb{R}^{\times}$ denotes the multiplicative group of real numbers.
My approach so far has been to first find a normal subgroup $N$ of $G$, and then try to construct an isomorphism between $G/N$ and $\mathbb{R}^{\times} \times \mathbb{R}^{\times}$.
One such normal subgroup is given by $N = \{\begin{pmatrix} d & 0\\ 0 & e \end{pmatrix}\ | d,e \in \mathbb{R}, de \neq 0\}$. It's easy to check that this is indeed a normal subgroup. But, the problem with this, and any other normal subgroup I end up constructing, is that now a typical element of $G/N$ looks like $gn = \begin{pmatrix} ad & ce\\ 0 & be \end{pmatrix}$, with $a,b,e,d \neq 0$, where $g, n$ are matrices in $G$ and $N$, respectively. This isn't promising in constructing an isomorphism between $G/N$ and $\mathbb{R}^{\times} \times \mathbb{R}^{\times}$. In particular, it has more than two "free" elements -- I was hoping for a matrix with only two "free" entries, each nonzero, so that I could just send that matrix to the pair of those elements, which will necessarily be an element of $\mathbb{R}^{\times} \times \mathbb{R}^{\times}$.
Is there a better way to approach this than trying to come up with a specific isomorphism? Or, is there a way I can guide myself to the right desired isomorphism?
Thanks!
Consider the group homomorphism $\phi$ that sends $$ \begin{pmatrix} a & b \\ 0 & c \end{pmatrix} \mapsto (a,c) \in \mathbb{R}^* \times \mathbb{R}^* $$
You should check this is a group homomorphism. The kernel of this homomorphism is the set $$ \begin{Bmatrix} \begin{pmatrix} 1 & b \\ 0 & 1 \end{pmatrix} : b \in \mathbb{R} \end{Bmatrix} $$
The kernel of a group homormorphism is a normal subgroup and by the first isomorphism theorem we have that $G/\ker\phi \cong \im \phi$. Thus, we have that the $\im\phi = \mathbb{R}^* \times \mathbb{R}^*$, so we get that $G/N \cong \mathbb{R}^* \times \mathbb{R}^*$ if $$ N = \begin{Bmatrix} \begin{pmatrix} 1 & b \\ 0 & 1 \end{pmatrix} : b \in \mathbb{R} \end{Bmatrix}. $$