Let $G$ be a finite connected group scheme over a field $k$, $H$ be a subgroup scheme of $G$. We could take for each $T/k$ the smallest normal subgroup of $G(T)$ containing $H(T)$. Thus we get a group functor. Let $N$ be the fppf sheafication of the group functor. Is $N$ representable?
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1$\begingroup$ Why do you prefer this functor, rather than considering for each $T$ the subgroup of $g \in G(T)$ such that $g$-conjugation on the $T$-group scheme $G_T$ carries the closed subgroup scheme $H_T$ into (and consequently isomorphically onto) itself? The alternative is what is called the "functorial normalizer" $N_G(H)$ and is often representable by a closed subgroup scheme of $G$ (e.g., it is for $G$ any finite type group scheme over a field). $\endgroup$– user28172Feb 6, 2013 at 23:16
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$\begingroup$ I'm not convinced that you have defined a functor. Why does it send $k$-morphisms $T \to T'$ to group homomorphisms? $\endgroup$– S. Carnahan ♦Feb 7, 2013 at 9:18
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$\begingroup$ @Carnahan, yes, you are completely right, thank you. I should ask for the largest normal subgroup of G(T) contained in H(T). $\endgroup$– unknownFeb 7, 2013 at 11:53
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$\begingroup$ @unknown: You are still making a mistake. Your modified version is generally not a subfunctor since normality is not functorial in group homomorphisms. So I ask again: please explain why you do not want to work with the notion of scheme-theoretic normalizer that is the one which "works" in SGA3. $\endgroup$– user28172Feb 18, 2013 at 4:56
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