Algebra and Geometry | Part IA, 2003

Let GG be the group of Möbius transformations of C{}\mathbb{C} \cup\{\infty\} and let X={α,β,γ}X=\{\alpha, \beta, \gamma\} be a set of three distinct points in C{}\mathbb{C} \cup\{\infty\}.

(i) Show that there exists a gGg \in G sending α\alpha to 0,β0, \beta to 1 , and γ\gamma to \infty.

(ii) Hence show that if H={gGgX=X}H=\{g \in G \mid g X=X\}, then HH is isomorphic to S3S_{3}, the symmetric group on 3 letters.

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