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 $PS{L}_2(ℂ)$. The isomorphism $\mathrm{\Psi }$ (of topological groups) is given by . (Here, the notation $[M]$ means the equivalence class $[M]=\{Mt\mid t\in ℂ\}$)

This mapping is:

Well-defined:

If 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 $z\mapsto \frac{a^{\prime }z+b^{\prime }}{c^{\prime }z+d^{\prime }}$.

A homomorphism:

Calculating the composition

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

we see that .

A monomorphism:

If , then it follows that $(a^{\prime },b^{\prime },c^{\prime },d^{\prime })=t(a,b,c,d)$,so that.

An epimorphism:

Any Möbius transformation $z\mapsto \frac{az+b}{cz+d}$ is the image .