Quotientes of afine group schemes Let $G=Spec(K[x_1,...,x_n])$ an afine group scheme and $H$ an subgroup scheme of $G$ then
-Can i say that $H$ is a afine group scheme, if not then when can I say it?
-How can I define the quotient (as a scheme) and  the resulting scheme is affine?
Thanks in advance.
 A: It might help if you give a precise definition of "subgroup scheme".  E.g. if you require that $H$ is a closed subgroup scheme, then $H$ will in particular be a closed subscheme of an affine scheme, and hence affine.
On the other hand, if $K = \mathbb C$ and $G = \mathbb G_a$ (otherwise known as $\mathbb G_a$, the additive group) then the morphisms $\mathbb Z \to G$
(where we think of $\mathbb Z$ as a discrete scheme; each point is just a copy of
Spec $\mathbb C$), given by sending each element $n \in \mathbb Z$ to the
corresponding closed point of $\mathbb G_a$, is a monomorphism, so that
certainly $\mathbb Z$ represents a group subfunctor of the functor of points
of $\mathbb G_a$ (so that it wouldn't be completely unreasonable to call
$\mathbb Z$ a subgroup scheme of $\mathbb G_a$), but $\mathbb Z$ is not an affine scheme (it has infinitely many
irreducible components).
In general, if $H$ is a closed subgroup scheme of the affine group scheme $G$,
then the quotient $G/H$ need not be affine.  E.g. if $G = GL_2$ and $H$
is the subgroup of upper triangular matrices, then $G/H = \mathbb P^1$, which is
not affine.
If $H$ is reductive, then $G/H$ is affine.  
(Maybe I should assume that
$K$ has char. zero, for safety.  Then here is a sketch of the proof: Consider a s.e.s. of coherent sheaves on $G/H$.  Taking global sections is the same
as pulling back to a s.e.s. on $G$, taking global sections
of these pull-backs over $G$, and then passing to $H$-invariants.  Now
pulling back to $G$ is exact, since $G \to G/H$ is flat, and passing
to global sections over $G$ is exact, since $G$ is affine.  Finally, passing
to $H$-invariants is exact, because $H$ is reductive.  Thus passing to global sections is an exact functor on coherent sheaves on $G/H$, and so by Serre's
cohomological criterion, $G/H$ is affine.)
If $H$ is a normal closed subgroup scheme of $G$, so that $G/H$ is again a group scheme, then
I guess $G/H$ is also necessarily affine.
