Find normal affine subgroup $N$ such that $G/N $ is an abelian variety ( $G=\mathbb{A}^1 \setminus 0 $)

Question

I'm reading Shafarevich Basic Algebraic Geometry. I read Chevalley Theorem. It asserts that every algebraic group $G$ has a normal subgroup $N$ such that $N$ is affine and $G/N$ is an abelian variety. I want to check Chevalley Theorem in one or two simple examples but I don't know how to do it.

For example: Let $X=\mathbb{A}^1 \setminus 0 $ this is an algebraic group endowed with the multiplication as the group operation. I want to find the affine subgroup $N$ but I don't know how =(

Answer 1

The commenter has already answered your question fully. However this piece of information is about a recent proof of Chevalley's theorem by M.Brion (with many more results). Please see the link below:

https://www-fourier.ujf-grenoble.fr/~mbrion/chennai.pdf