Why $G\to G/H$ is faithfully flat? Some questions about algebraic groups.
Let $G$ be an affine algebraic group over algebraically closed field $k$. 
Questions: Let $H$ be a closed subgroup of $G$, then (as I learnt from some paper) the quotient $\pi\colon G\to G/H$ is faithfully flat, why? reference? When it is locally trivial, especially when $G$ is linear algebraic group?
Thank you very much!
 A: If you are really looking for a reference, here is one: Groupes algébriques, Demazure-Gabriel, Chapter III, §3, Proposition 2.5 p. 328. Note that the fact that the base is a field is not important. It might be any scheme. The important assumptions are 


*

*the subgroup $H$ has to be flat over the base (which of course holds if it is smooth, which always hold over a field in char 0),

*the quotient $G/H$ (defined as the fppf sheafification of the presheaf quotient) has to be a scheme (i.e. representable). This holds when $G$ is affine over a base field, by theorem 5.4, also in Ch. III, §3.


P.S. I am aware that this book is in french, and that the notation used in it makes it quite hard to browse through without spending too much time. Note, however, that it contains an index for the notation at the end. It is a very thorough reference for this type of questions.
P.P.S. As for the "locally trivial" part of the question, do you mean Zariski locally trivial (i.e. there is a Zariski open subset U of G/H over which as a scheme, the morphism becomes $U\times H \to U$)?
A: [Assume the group schemes are reduced. [Görtz-Wedhorn], Theorem 14.5 gives generic flatness, and $\pi: G \to G/H$ is a homogeneous space ([Borel], LAG §6). For surjectivity, see [Borel], LAG §6.
