Past exam of the mathematics course of the University of Cambridge / 2008 / ia / Paper 3 / 7E / iii / Solution

The assertion is true over the complex numbers. It suffices to generate every nonzero scaling using translations and $I(z)=1/z$, since these together with scalings already generate the Möbius group.

For $s\ne 0$, compose the maps rightmost first. Away from the intermediate poles, the successive values areHenceFor every $\lambda \ne 0$, choose a complex square root $s$ with $s^2=-\lambda$. This gives $D_{\lambda }$ and proves Möbius scaling generated by translations and reciprocal inversion. Equality of the rational maps extends the identity to all points of the Riemann sphere, including the intermediate exceptional values.