We have the following Group $\mathbb{R}^{\times}$ acting on $\mathbb{R²}$ with
\begin{matrix}
\mathbb{R^{\times}} \times \mathbb{R}^2 \rightarrow \mathbb{R}^2 \\
(t,(x,y)) \mapsto (tx, \frac{y}{t})
\end{matrix}
How to compute the (i) orbits and (ii) stabilizers ?
(i) I would say that the orbit of such an element $x=(a,b)$ is
\begin{equation}
\mathbb{R}^{\times}.x=\{(ta, \frac{b}{t}) \vert t \in \mathbb{R}^{\times} \}
\end{equation}
or given by the linear function
\begin{equation}
f(t) = \frac{ta – a}{b – \frac{b}{t}}, \text{ where } t \in \mathbb{R}\setminus\{ 0 \}
\end{equation}
(ii) The stabilizer $\mathbb{R}^{\times}_x = \{1\}$ because $(x,y) = (tx, \frac{y}{t})$ only if $t=1$.
This was my approach and i would appreciate some help to work this out.
Best Answer
Simpler hint for (i):
If $x\ne 0$ and $y\ne 0$ observe that, for any $t\ne 0$, we have $$tx\cdot\frac yt=xy,$$ so the orbit of $(x,y)$ is a well-known curve.
Next, consider separately the cases $x=0,\:y\ne 0$, $x\ne 0, \: y=0$, $(x,y)=(0,0)$.