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Let $p$ be a prime number and $R$ be a Noetherian local ring of characteristic $p$ with residue field $k$. Let $G$ be a finite etale subgroup scheme over $R$ of order $p$. Suppose that the etale $k$-group scheme $G_{s}:=G \times_{S}$Spec$(k)$ is isomorphic to the constant group $\mathbb{Z} / p\mathbb{Z}$. Then is it true that $G$ is itself isomorphic to $\mathbb{Z}/p\mathbb{Z}$ as $R$-groups? (Note that $R$ may not be Henselian).


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up vote 4 down vote accepted

Take $p=3$, and take an étale double cover $X\to S=\mathrm{Spec}(R)$. Let $E$ be a copy of $S$, and $G=E\coprod X$. It is easy to see that there is a unique $S$-group scheme structure on $G$ with $E$ as unit section. Now there are plenty of examples where $X$ is nontrivial but is trivial over the closed point.

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