Greeting, i'm confused about the following definition of a normal subgroup of a group scheme
What does the writer mean by the map $(h,g)\rightarrow ghg^{-1}$? Knowing that we work with Group schemes and not the regular groups? Why is this equivalent to the fact that $H(R)$ is a normal subgroup of $G(R)$ for all $k$-algebras $R$?
I bet if you combined your two questions you might be able to answer it yourself.
The map $H\times G\to G$ is the map which, on $R$-points for a $k$-algebra $R$, sends $(h,g)\in H(R)\times G(R)$ to $hgh^{-1}$ in $G(R)$. You can answer your second question then I suspect.