Past exam of the metric spaces and complex analysis course of the University of Oxford / 2014 / 6 / b

When all three points are finite, useThe distinctness assumptions ensure that it is a nonconstant Möbius transformation, and direct substitution gives the three required values. If one of the specified points is infinity, the limiting formulas areThese cover every case because the points are distinct.

For uniqueness, if $g$ and $\tilde{g}$ have the same three images, the Möbius map $\tilde{g}\circ g^{-1}$ fixes $0,1,\infty$. Fixing zero and infinity makes it $z\mapsto cz$ with $c\ne 0$; fixing one gives $c=1$. Therefore the normalized map is unique. This is the sharp three-transitivity of Möbius transformations.

Let $G$ be the subgroup preserving the three-point set. Its action gives a group homomorphism $\Phi :G\to S_3$. If $\Phi (h)$ is the identity permutation, then $h$ fixes all three points, so part (i) forces $h$ to be the identity Möbius map. Thus the kernel is trivial and $\Phi$ is injective.

For any permutation $\sigma$, part (i) gives a normalization $g_{\sigma }$ sending the ordered triple $(z_{\sigma (1)},z_{\sigma (2)},z_{\sigma (3)})$ to $(0,1,\infty )$. If $g$ normalizes the original triple, then $h_{\sigma }=g_{\sigma }^{-1}\circ g$ sends $z_j$ to $z_{\sigma (j)}$. Hence every permutation is attained, so $\Phi$ is surjective.This is the setwise stabilizer; its six maps need not fix each point individually.

Choose a Möbius map $T$ with $T(z_1)=0$ and $T(z_2)=\infty$, for example by normalizing the ordered triple $(z_1,z_3,z_2)$. Conjugation $h\mapsto T\circ h\circ T^{-1}$ takes the requested subgroup to the subgroup fixing zero and infinity individually.

Such a transformation must be $z\mapsto cz$ with $c\in {\mathbb{C}}^{\ast }$: the zero condition removes the constant numerator term and the infinity condition removes the nonconstant denominator term. Conversely every such dilation is Möbius and fixes the two points. Composition multiplies its nonzero parameter.This identifies the Möbius pointwise stabilizer of two points. Allowing an interchange of the two fixed points would give a larger group, but those interchanges are excluded by the stated equalities.