= Möbius pointwise stabilizer of two points
{c}
{title2=$G_{(p,q)}\cong\mathbb C^*$}
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 $z\mapsto\lambda z$, $\lambda\ne0$. 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 $z\mapsto\lambda z/[1+(\lambda-1)z]$.
Back to article page
