Show that the usual action of SL2(Z) (the group of 2×2 integer matrices with determinant 1 ) on R2 induces a transitive action of SL2(Z) on S .

Question

Let $S$ be the set of all lines in $\mathbf{R}^{2}$ that pass through the origin and have either rational or infinite slope. Show that the usual action of $SL_{2}(\mathbf{Z})$ (the group of $2 \times 2$ integer matrices with determinant $1$) on $\mathbf{R}^{2}$ induces a transitive action of $SL_{2}(\mathbf{Z})$ on $S$.

My Attempt: We need to construct $A\in SL_{2}(\mathbf{Z})$ such that $A\begin{pmatrix} 1\\p \end{pmatrix}= \begin{pmatrix} 1\\q \end{pmatrix}$ for distinct $p,q \in \mathbf{Q}$. But I am not able to construct $A$, give some Hints.

Answer 1

Put some of these facts together.

• $\mathrm{SL}_2\mathbb{Z}\curvearrowright S$ transitively iff $S$ is an orbit, say of $\mathbb{R}\times\{0\}=\mathrm{span}\{(\begin{smallmatrix}1\\0\end{smallmatrix})\}$.
• A line of slope $m=\frac{a}{b}$ is the span of $(\begin{smallmatrix}a\\b\end{smallmatrix})$.
• Every rational number $\frac{a}{b}$ has a reduced form where $\gcd(a,b)=1$.
• Bezout's theorem for pairs of coprime integers.