Lagrange's theorem for a family 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 family of non-commutative k-group schemes is killed by its order. This is achieved via a classification result which rely on the explicit 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.