Affine group of the complex line
= Affine group of the complex line
{c}
{title2=$\operatorname{Aff}(\mathbb C)$}
The affine group of the complex line consists of the maps
$$
f_{a,b}(z)=az+b,
\qquad a\in\mathbb C^\times,\quad b\in\mathbb C.
$$
Composition identifies it with the <semidirect product> $(\mathbb C,+)\rtimes\mathbb C^\times$, where $a$ acts on a translation coordinate by multiplication.