Suppose $X$ is a finite flat group scheme over $\mathbb Z$, killed by a prime number $p$ and such that there exists an extension as finite flat group schemes defined over $\mathbb Z$:
$$0\to \mathbb{Z}/p\mathbb{Z}\to X \to \mu\_p \to 1.$$



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> **Question:** Can we conclude that $X\cong \mathbb{Z}/p\mathbb{Z}\times \mu\_p$ over $\mathbb{Z}$?
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I know that the answer to this question is negative in general if you consider it over $\mathbb Q$, since you can take $X=E[7]$, the group scheme of $p$-torsion points of an elliptic curve with a $7$-torsion point defined over $\mathbb Q$, since we have such elliptic curves but no such curve with $E[7]\cong \mathbb{Z}/7\mathbb{Z}\times \mu\_7$. Of course you can find examples easily for $p=2,3,5$, and probably for infinitely many prime numbers.


On the other hand, over the finite field $\mathbb{F}\_p$ the answer is positive, since $\mu\_p$ is connected and $\mathbb{Z}/p\mathbb{Z}$ étale, and one could use the connected-étale exact sequence of $X$ to get an splitting of the exact sequence above.


If the answer to the question is affirmative, I will be also interested for what other ring of integers the result is true. I suspect it should be related to the fact that $\mathbb{Q}$ has no unramified extensions.



Finite flat group schemes over $\mathbb{Z}$ that are extensions of $\mu_p$ by $\mathbb{Z}/p\mathbb{Z}$
It is proved in step 3 and 4 of section 3.4.3 in J-M. Fontaine. Il n’y a pas de variété abélienne sur Z. Invent. Math., 81(3):515– 538, 198 (using the ramification bound in that paper) that:


For $E=\mathbb Q$ and $\mathbb Q(\sqrt{-1})$, $\mathbb Q(\sqrt{-3})$, in the category of finite flat group schemes over $O\_E$ killed by $p$, there is no non-trivial extension of $\mu\_p$ by $\mathbb Z / p \mathbb Z$.


