(Updated)
I have looked the draft of Ch4 of the book "Abelian Varieties" by Gerard van der Geer and Ben Moonen. It looks like in order to see the group scheme structure on G/H, one should consider the fppf quotient. It is eaiser to see the group scheme structure on the fppf quotient. And one can prove that the fppf quotient is equal to the category quotient. Is this the standard way? ( In fact, I am curious if we need the notion of grothendieck topology to see the group scheme structure on G/H)
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Let me just mention the original question for this topic which is about the quotient of a scheme by a finite group scheme action. In SGA 3, the (general) definition is as following: Consider a diagram
$$ X_1 { \xrightarrow[]{d_0} \atop \xrightarrow[d_1]{} } X_0 \xrightarrow{ \ p \ } Y$$
We call $(Y,p)$ is a quotient if $p \circ d_0 = p \circ d_1$ and for any $q: X_0 \rightarrow Z$ such that $q \circ d_0 = q \circ d_1$, there exists a unique $r: Y \rightarrow Z$ such that $q = r \circ p$. The existence of the quotient $Y$ is equivalent to the representability of the functor $K: T \rightarrow K(T) $, i.e $K=\mathrm{Hom}(Y,-)$, here $K(T)$ is the kernel of
$$ \mathrm{Hom}(X_0, T) { \xrightarrow[]{T(d_0)} \atop \xrightarrow[T(d_1)]{} } \mathrm{Hom}(X_1, T) $$
In SGA 3, it's proved that the quotient exists in some case.
On the other hand, on wikipedia(group scheme), it's written that:
"For a subgroup scheme H of a group scheme G, the functor that takes an S-scheme T to G(T)/H(T) is in general not a sheaf, and even its sheafification is in general not representable as a scheme. However, if H is finite, flat, and closed in G, then the quotient is representable, and admits a canonical left G-action by translation. If the restriction of this action to H is trivial, then H is said to be normal, and the quotient scheme admits a natural group law. Representability holds in many other cases, such as when H is closed in G and both are affine.[1]"
It looks like that the two definitons of quotient are different. The first one considers morphisms to an object $T$ and the second definition considers morphisms from $T$. The first definition seems more natural to me for the quotient.
My question is : are these two definitions equivalent, under the following assumptions:
$X_0 = G$ is a group scheme, $X_1 = H \times G$ for a finite closed subgroup scheme $H$, $d_0 = m$ being the induced morphism from the multiplication and $d_1$ is the second projection.
ps: When I tried to figure out how to give a multiplication on the quotient $G/H$ (of course, one needs the condition "normal"), I have the first definition of quotient in mind, and can't see why "...and admits a canonical left G-action by translation". Using the second definition, it is easy to see it.
