Algebra and Geometry | Part IA, 2006

Show that every Möbius transformation can be expressed as a composition of maps of the forms: S1(z)=z+α,S2(z)=λzS_{1}(z)=z+\alpha, S_{2}(z)=\lambda z and S3(z)=1/zS_{3}(z)=1 / z, where α,λC\alpha, \lambda \in \mathbb{C}.

Show that if z1,z2,z3z_{1}, z_{2}, z_{3} and w1,w2,w3w_{1}, w_{2}, w_{3} are two triples of distinct points in C{}\mathbb{C} \cup\{\infty\}, there exists a unique Möbius transformation that takes zjz_{j} to wj(j=1,2,3)w_{j}(j=1,2,3).

Let GG be the group of those Möbius transformations which map the set {0,1,}\{0,1, \infty\} to itself. Find all the elements of GG. To which standard group is GG isomorphic?

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