These factor groups are isomorphic to which group

Question

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?

1. $G/N$ is isomorphic to $\mathbb{R}$ under addition.

2. $G/N$ is isomorphic to $\{a \in\mathbb{R}: a>0\}$ under multiplication.

3. 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.

Answer 1

Let $\Bbb R^*$ be a set $\{a \in\mathbb{R}: a>0\}$. It is easy to check that $(\Bbb R^*,\cdot )$ is a group and a map $f:(\Bbb R^*,\cdot)\to (\Bbb R,+)$, $x\mapsto \log(x)$ is an isomorphism of groups. So (1) and (2) are equivalent.

Now, as lhf pointed, consider a map $h: G\to \Bbb R^*$, $\begin{pmatrix}a&b\\ 0&a^{-1} \end{pmatrix} \mapsto a$. An equality $$\begin{pmatrix}a_1 & b_1\\ 0 & a_1^{-1} \end{pmatrix}\begin{pmatrix}a_2 & b_2\\ 0 & a_2^{-1} \end{pmatrix}=\begin{pmatrix}a_1a_2 & a_1b_2+a_2^{-1}b_1\\ 0 & a_1^{-1}a_2^{-1}\end{pmatrix}$$ valid for all $a_1, a_2\in \Bbb R^*$ and $b_1, b_2\in \Bbb R$ shows that $h$ is a surjective homomorphism from $(G,\cdot)$ to $\mathbb (\Bbb R^*,\cdot)$. Clearly, $\ker h=N$. Thus we proved (2).

To show (3), it suffices to pick an arbitrary proper normal subgroup $H’$ of $(\Bbb R^*,\cdot)$ which properly contains $1$ (for instance, $H’=\{e^n:n\in\Bbb Z\}$) and put $N’=h^{-1}(H')$.

Answer 2

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 $\mathbb R^*_+ \to \mathbb R$.