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Some questions about algebraic groups.

Let $G$ be an affine algebraic group over algebraically closed field $k$. Questions: 1. $G$ is faithfully flat since it is defined over field? 2. Let $H$ be a closed subgroup of $G$, then (as I learnt from some paper) the map $\pi\colon G\to G/H$ is faithfully flat, why? reference? When it is locally trivial, especially when $G$ is linear algebraic group?

Thank you very much!

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    Sorry, I wasn't online over the weekend. I will look it up this evening, than send you the reference.2012-07-23

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