Algebra and Geometry | Part IA, 2001

Show that the set of Möbius transformations of the extended complex plane C{}\mathbb{C} \cup\{\infty\} form a group. Show further that an arbitrary Möbius transformation can be expressed as the composition of maps of the form

f(z)=z+a,g(z)=kz and h(z)=1/zf(z)=z+a, \quad g(z)=k z \text { and } h(z)=1 / z

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