Let $G$ be an affine algebraic group over $\mathbb{C}$. According to SGA3, any closed normal subgroup $N$ is representable by an affine algebraic group, as is the quotient $G/N$.


These statements are valid in the fpqc topology: that is, they are true when considering algebraic groups as group objects in the category of sheaves on the big (affine) fpqc site of $\mathrm{Spec}\;\mathbb{C}$. My question is:



> 
> Can I lift this to the étale topology?
> 
> 
> 


Even if this cannot be done in general, is there a know set of conditions on $G$ (maybe on $N$ too) that guarantees that the map $G \to G/N$ is also a quotient in the étale topology?



Quotient of algebraic groups in the étale topology
The map $G\to G/N$ is always a quotient in the étale topology. Since you are in characteristic $0$, the group scheme $N$ is smooth. Since $G \to G/N$ is an $N$-torsor, because the action of $N$ on $G$ is free, and an fpqc torsor under a smooth group is also and étale torsor, because a smooth map always has local sections in the étale topology.


