 Solution

Choose algebra generators $f_1,\dots ,f_r$ of the finitely generated coordinate ring $k[G]$. The right regular action is locally finite: writingshows that every right translate of $f_i$ lies in the finite span of the $f_{ij}$. Hence all the $f_i$ lie in some finite-dimensional translation-stable subspace $V\subseteq k[G]$.

This gives a rational representation $\rho :G\to \mathrm{GL}(V)$. If $g\in \ker \rho$, then $f_i(xg)=f_i(x)$ for every $i$ and $x$. Setting $x=e$ gives $f_i(g)=f_i(e)$ for every algebra generator, and therefore for every regular function on $G$. Regular functions separate closed points of an affine variety, so $g=e$. Thus $\rho$ is a faithful representation of an affine algebraic group.

Solved by gpt-5.6-sol high.