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 $\Psi: \left[\smfour{a}{b}{c}{d}\right] \mapsto \mobius{z}{a}{b}{c}{d}$ (Here, the notation $[M]$ means the equivalence class $[M] = \{ Mt \mid t \in \mathbb{C} \}$

This mapping is:

Well-defined:

If $\left[\smfour{a}{b}{c}{d}\right]=\left[\smfour{a'}{b'}{c'}{d'}\right]$ then $(a',b',c',d')=t(a,b,c,d)$ for some $t$ so $z\mapsto\mobius{z}{a}{b}{c}{d}$ is the same transformation as $z\mapsto\mobius{z}{a'}{b'}{c'}{d'}$

A homomorphism:

Calculating the composition $$ \left.\mobius{z}{a}{b}{c}{d}\right|_{z=\mobius{w}{e}{f}{g}{h}} = \frac{a\mobius{w}{e}{f}{g}{h}+b}{c\mobius{w}{e}{f}{g}{h}+d} = \frac{(ae+bg)w+(af+bh)}{(ce+dg)w+(cf+dh)} $$ we see that $\Psi\left(\left[\smfour{a}{b}{c}{d}\right]\right)\cdot \Psi\left(\left[\smfour{e}{f}{g}{h}\right]\right) = \Psi\left(\left[\smfour{a}{b}{c}{d}\right]\cdot \left[\smfour{e}{f}{g}{h}\right]\right)$

A monomorphism:

If $\Psi\left(\left[\smfour{a}{b}{c}{d}\right]\right)= \Psi\left(\left[\smfour{a'}{b'}{c'}{d'}\right]\right)$ then it follows that $(a',b',c',d')=t(a,b,c,d)$ so that $\left[\smfour{a}{b}{c}{d}\right]= \left[\smfour{a'}{b'}{c'}{d'}\right]$

An epimorphism:

Any Möbius transformation $z\mapsto\mobius{z}{a}{b}{c}{d}$ is the image $\Psi\left(\left[\smfour{a}{b}{c}{d}\right]\right)$