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!