2
$\begingroup$

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.

$\endgroup$
3
$\begingroup$

Hint: try with $$ \begin{bmatrix} a & b \\ 0 & a^{-1} \end{bmatrix} \mapsto a $$ as an homomorphism from $G$ to $\mathbb{R}^+$.

$\endgroup$
  • $\begingroup$ but why is the group isomorphic to $(\mathbb R,+)$ $\endgroup$ – Learnmore Nov 16 '15 at 12:59
  • $\begingroup$ $\mathbb{R}$ is isomorphic to $\mathbb{R}^+$ via the exponential function $\endgroup$ – Ottavio Bartenor Nov 16 '15 at 14:18
  • $\begingroup$ Yes you are right ;thank you very much+1 $\endgroup$ – Learnmore Nov 16 '15 at 14:35
  • $\begingroup$ You're welcome! $\endgroup$ – Ottavio Bartenor Nov 16 '15 at 14:39
2
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ Not sure really what are you talking about? $\endgroup$ – Learnmore Nov 16 '15 at 6:57
  • $\begingroup$ sorry; small mistake was done here. $\endgroup$ – Groups Nov 16 '15 at 7:13

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.