Möbius pointwise stabilizer of two points ($G_{(p,q)}\cong {\mathbb{C}}^{\ast }$)

For two distinct points of the Riemann sphere, conjugate by a Möbius transformation sending them to zero and infinity. A transformation fixing those two points has the form $z\mapsto \lambda z$, $\lambda \ne 0$. Composition multiplies the parameter, so the pointwise stabilizer is isomorphic to the complex multiplicative group. For zero and one, conjugation by $z/(1-z)$ gives $z\mapsto \lambda z/[1+(\lambda -1)z]$.