Show that every Möbius map may be expressed as a composition of maps of the form $z \mapsto z+a, z \mapsto \lambda z$ and $z \mapsto 1 / z$ (where $a$ and $\lambda$ are complex numbers).

Which of the following statements are true and which are false? Justify your answers.

(i) Every Möbius map that fixes $\infty$ may be expressed as a composition of maps of the form $z \mapsto z+a$ and $z \mapsto \lambda z$ (where $a$ and $\lambda$ are complex numbers).

(ii) Every Möbius map that fixes 0 may be expressed as a composition of maps of the form $z \mapsto \lambda z$ and $z \mapsto 1 / z$ (where $\lambda$ is a complex number).

(iii) Every Möbius map may be expressed as a composition of maps of the form $z \mapsto z+a$ and $z \mapsto 1 / z$ (where $a$ is a complex number).