# How to show that the quotient group is isomorphic to $(\mathbb R,+)$

Let $$G= \begin{Bmatrix} \begin{bmatrix} a & b \\ 0 & a^{-1} \end{bmatrix} \colon a,b\in \mathbb R; a>0 \end{Bmatrix} \mbox{ and } N= \begin{Bmatrix} \begin{bmatrix} 1 & b \\ 0 & 1 \end{bmatrix} \colon b\in \mathbb{R} \end{Bmatrix}$$

Prove that $G/N$ is isomorphic to $(\mathbb R,+)$ and $G/N$ is isomorphic to $(\mathbb R^{+},*)$.

I have to get a onto homomorphism from $G$ to $\mathbb R$ whose kernel is $N$ .I am failing to do so.

Any hint would suffice.

Hint: try with $$\begin{bmatrix} a & b \\ 0 & a^{-1} \end{bmatrix} \mapsto a$$ as an homomorphism from $G$ to $\mathbb{R}^+$.
• but why is the group isomorphic to $(\mathbb R,+)$ – Learnmore Nov 16 '15 at 12:59
• $\mathbb{R}$ is isomorphic to $\mathbb{R}^+$ via the exponential function – Ottavio Bartenor Nov 16 '15 at 14:18
Consider the natural map: $G\rightarrow G$, given by $$\begin{bmatrix} a & b \\ 0 & a^{-1} \end{bmatrix} \mapsto \begin{bmatrix} a & 0 \\ 0 & a^{-1} \end{bmatrix}.$$ Is it a homomorphism? What is kernel and image?