Affine group of the complex line ($\mathrm{Aff}(\mathbb{C})$)

The affine group of the complex line consists of the mapsComposition identifies it with the semidirect product $(\mathbb{C},+)⋊{\mathbb{C}}^{\times }$, where $a$ acts on a translation coordinate by multiplication.

For the Affine group of the complex line,Indeed, the multiplier map $f_{a,b}\mapsto a$ has the displayed translation group as its kernel and abelian image, whileproduces every translation.