# Isomorphism of quotient group

Let $G=\left\{\begin{bmatrix}a&b\\0&a^{-1}\end{bmatrix}:a,b\in \mathbb{R}, a>0\right\}$ and $N=\left\{\begin{bmatrix}1&b\\0&1\end{bmatrix}:b\in \mathbb{R}\right\}$

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 $N$

4. $N$ is isomorphic to $\mathbb{R}$ under addition.

Consider $\eta:G\rightarrow N$ with $\begin{bmatrix}a&b\\0&a^{-1}\end{bmatrix}\mapsto a$ this is clealy a homomorphism with kernel $N$ so second option is correct..

Consider $\eta:G\rightarrow N$ with $\begin{bmatrix}a&b\\0&a^{-1}\end{bmatrix}\mapsto b$ this is clealy a homomorphism which is bijective so fourth option is correct..

I am not able to decide whether other options are true or false..

• The groups specified in 1 and 2 are isomorphic, so 1 is true if and only if 2 is true. 3 is also true (for example, restrict $a$ to rational numbers). It looks as though they are all true. – Derek Holt Mar 11 '16 at 9:12
• I guess $x\mapsto e^x$ is the isomorphism from $\mathbb{R},+$ to $\{a>0\}$.. AM i correct? @DerekHolt – user312648 Mar 11 '16 at 9:38
• Yes that's right! – Derek Holt Mar 11 '16 at 13:00
• @DerekHolt : Thanks :) – user312648 Mar 11 '16 at 13:10