Grothendieck’s group-scheme question

Grothendieck asked in the 1960s whether every finite locally free group scheme is annihilated by multiplication by its order, without assuming commutativity. The unrestricted question is not settled.

Known results

• Grothendieck, SGA 3: true over fields and, more generally, reduced base schemes.
• Deligne, around 1970: true for commutative finite flat group schemes.
• Schoof, 2001: true over local Artin rings with residue characteristic $p$ and $\mathfrak m_R^p=p\mathfrak m_R=0$.
• Tate–Oort: group schemes of prime order $p$ are killed by $p$.

November 18, 2024 partial result

On November 18, 2024, the preprint Lagrange’s theorem for a family of finite flat group schemes over local Artin rings claimed the result for finite flat deformations of specified noncommutative group schemes $G_\lambda$ with $\lambda\in\{1,\ldots,p^{m-1}\}$. It explicitly says the case $G_0=\alpha_p\times\mu_{p^m}$ remains problematic and that the full problem is open.

Current status (as of August 2026): established for commutative group schemes, reduced bases, and several noncommutative families, while the general noncommutative case remains open.