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See SGA3, Exp. VI$_{\rm{A}}$, Theorem 3.2(iv): for any group scheme $G$ locally of finite type over a field $k$ and any closed subgroup scheme $H$ there exists a locally finite type $k$-scheme $G/H$ equipped with a faithfully flat $k$-morphism $G\rightarrow G/H$ invariant under the right $H$-action such that $G×H→G\times_{G/H}G$ is an isomorphism. Thus, this has all "good" quotient properties you could ever want (for maps out, maps in, geometric points, etc.), by descent theory. (The result also holds over any artinian local ring, assuming $H$ is flat.)

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