Lagrange’s theorem for a class of finite flat group schemes over local Artin rings

Abstract

Let $R$ be a local Artin ring with residue field $k$ of positive characteristic. We prove that every finite flat group scheme over $R$ whose special fiber belongs to a certain explicit class of non-commutative $k$-group schemes is killed by its order. This is achieved via a classification result which relies on the study of the infinitesimal deformation theory for such non-commutative $k$-group schemes. The main result answers positively in a new case a question of Grothendieck in SGA 3 on whether all finite flat group schemes are killed by their order, and improves the currently best known result due to Schoof.