I am asking a question from Abstract Algebra Assignment in which I am having a trouble.
Let $$G=\left\{\begin{pmatrix}a&b\\0&a^{-1} \end{pmatrix}: a,b\in\mathbb{R} , a>0\right\}$$ and $$N=\left\{\begin{pmatrix}1&b\\ 0&1 \end{pmatrix}: b\in\mathbb{R}\right\}.$$ Then which of the following are true?
$G/N$ is isomorphic to $\mathbb{R}$ under addition.
$G/N$ is isomorphic to $\{a \in\mathbb{R}: a>0\}$ under multiplication.
There is a proper normal subgroup $N'$ of $G$ which properly contains $N$.
For option 1,2 I am really confused what $G/N$ will look like although I know that now multiplication and addition will be Mod $N$. So, I would really like to work out 1,2 myself if one can just tell me structure of $G/N$.
For Option 3, I need complete guidance as I have no clue for this.
I shall be really thankful for your help.
Best Answer
Hint: Prove that $\begin{pmatrix}a&b\\ 0&a^{-1} \end{pmatrix} \mapsto a$ is a surjective homomorphism $G \to \mathbb R^*_+$ with kernel $N$. Then compose with an isomorphism $\mathbb R^*_+ \to \mathbb R$.