# isomorphism of the group PSL_2(C) with the group of Möbius transformations

We identify the group $G$ of Möbius transformations  with the projective special linear group  $PSL_{2}(\mathbb{C})$. The isomorphism         $\Psi$ (of topological groups) is given by $\Psi:\left[{\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)}\right]\mapsto{\frac{az+b}{cz+d}}$. (Here, the notation $[M]$ means the equivalence class   $[M]=\{Mt\mid t\in\mathbb{C}\}$)

This mapping is:

Well-defined:

If $\left[{\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)}\right]=\left[{\left(\begin{smallmatrix}{a^{% \prime}}&{b^{\prime}}\\ {c^{\prime}}&{d^{\prime}}\end{smallmatrix}\right)}\right]$ then $(a^{\prime},b^{\prime},c^{\prime},d^{\prime})=t(a,b,c,d)$ for some $t$, so $z\mapsto{\frac{az+b}{cz+d}}$ is the same transformation as $z\mapsto{\frac{a^{\prime}z+b^{\prime}}{c^{\prime}z+d^{\prime}}}$.

Calculating the composition

 $\left.{\frac{az+b}{cz+d}}\right|_{z={\frac{ew+f}{gw+h}}}=\frac{a{\frac{ew+f}{% gw+h}}+b}{c{\frac{ew+f}{gw+h}}+d}=\frac{(ae+bg)w+(af+bh)}{(ce+dg)w+(cf+dh)}$

we see that $\Psi\left(\left[{\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)}\right]\right)\cdot\Psi\left(\left[{\left(% \begin{smallmatrix}{e}&{f}\\ {g}&{h}\end{smallmatrix}\right)}\right]\right)=\Psi\left(\left[{\left(\begin{% smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)}\right]\cdot\left[{\left(\begin{smallmatrix}{e% }&{f}\\ {g}&{h}\end{smallmatrix}\right)}\right]\right)$.

If $\Psi\left(\left[{\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)}\right]\right)=\Psi\left(\left[{\left(\begin{% smallmatrix}{a^{\prime}}&{b^{\prime}}\\ {c^{\prime}}&{d^{\prime}}\end{smallmatrix}\right)}\right]\right)$, then it follows that $(a^{\prime},b^{\prime},c^{\prime},d^{\prime})=t(a,b,c,d)$, so that $\left[{\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)}\right]=\left[{\left(\begin{smallmatrix}{a^{% \prime}}&{b^{\prime}}\\ {c^{\prime}}&{d^{\prime}}\end{smallmatrix}\right)}\right]$.

Any Möbius transformation $z\mapsto{\frac{az+b}{cz+d}}$ is the image $\Psi\left(\left[{\left(\begin{smallmatrix}{a}&{b}\\ {c}&{d}\end{smallmatrix}\right)}\right]\right)$.

Title isomorphism of the group PSL_2(C) with the group of Möbius transformations IsomorphismOfTheGroupPSL2CWithTheGroupOfMobiusTransformations 2013-03-22 12:43:30 2013-03-22 12:43:30 rspuzio (6075) rspuzio (6075) 9 rspuzio (6075) Result msc 57S25