Prove that determinant is a morphism from $PGL_2(\Bbb F_p)$ to $(\Bbb F_p^*/\Bbb F_p^{*2})$ where $PGL_2(\Bbb F_p) = GL_2(\Bbb F_p)/\Bbb F_p^*.I_2$.
What tools do we have from group theory to tackle this question? If we want to avoid heavy calculations, could we just say that determinant of scalar matrices are squares in $F_p^*$?.
More generally, if a morphism $f$ sends a group $G$ to $G'$ and a subgroup $H$ of $G$ to $H'$ a subgroup of $G'$, could we say that f induces a morphism $G/H\to G'/H'$?
Thank you for your help.