Past exam of the mathematics course of the University of Cambridge / 2008 / ia / Paper 3 / 7E (Groups)

Write a Möbius transformation as $M(z)=(az+b)/(cz+d)$ with $ad-bc\ne 0$, interpreted on the Riemann sphere. Let $T_u(z)=z+u$, $D_{\lambda }(z)=\lambda z$ with $\lambda \ne 0$, and $I(z)=1/z$.

If $c=0$, then $a,d\ne 0$ and $M(z)=(a/d)z+b/d=T_{b/d}\circ D_{a/d}(z)$. If $c\ne 0$, algebraic division givesConsequentlyThe scaling coefficient is nonzero because $ad-bc\ne 0$. Thus translations, nonzero complex scalings and reciprocal inversion generate every Möbius map. The formulas extend over their poles and infinity by the sphere interpretation.

The assertion is true. A Möbius map fixes infinity if and only if $c=0$: if $c\ne 0$, its value there is the finite number $a/c$. For $c=0$, the decomposition above uses only a nonzero complex scaling and a translation. These maps form the Affine subgroup of the Möbius group.

The assertion is false. Let $D_{\lambda }(z)=\lambda z$ and $I(z)=1/z$. Their relations areMoving each inversion to the right reduces every word to $D_{\lambda }{I}^{ϵ}$ with $ϵ=0$ or $1$. Thus the subgroup generated by complex scalings and reciprocal inversion consists only of $z\mapsto \lambda z$ and $z\mapsto \lambda /z$. Of these, only the first form fixes zero.

The Möbius transformationfixes zero but is not a scaling: it sends infinity to $1$ instead of infinity. It therefore cannot belong to the subgroup generated by scalings and reciprocal inversion.

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.