Past exam of the mathematics course of the University of Cambridge / 2026 / ia / Paper 3 / 6D / ii / Solution

The Möbius group consists of maps $z\mapsto (az+b)/(cz+d)$ with $ad-bc\ne 0$, including the usual action on $\infty$. Scalar matrices act trivially and are the whole kernel, so $M\cong G{L}_2(\mathbb{C})/({\mathbb{C}}^{\ast }I)$.

Solved by gpt-5.6-sol high.