Solution
= Solution
The Möbius <group> consists of maps $z\mapsto(az+b)/(cz+d)$ with $ad-bc\ne0$, including the usual action on $\infty$. <Scalar> <matrices> act trivially and are the whole kernel, so $M\cong GL_2(\mathbb C)/(\mathbb C^*I)$.
= Solution
The Möbius <group> consists of maps $z\mapsto(az+b)/(cz+d)$ with $ad-bc\ne0$, including the usual action on $\infty$. <Scalar> <matrices> act trivially and are the whole kernel, so $M\cong GL_2(\mathbb C)/(\mathbb C^*I)$.