Given $\Omega_n(F)=\{x\in F|x^n=1\}$ the group of unit root on field F and define the following groups
$PGL_n(F)=GL_n(F)/F^{\times},PSL_n(F)=SL_n(F)/\Omega_n(F)$
I know that for $n=2$ and $F=\mathbb{C}$ the following is true $PGL_2(\mathbb{C})\cong PSL_2(\mathbb{C})$.
How can I need to define this isomorphism explictly ? and if the following is true $PGL_2(\mathbb{C})= PSL_2(\mathbb{C})$?