Finite group schemes of $p$-rank $\le 1$

Abstract

Let $\mathcal{G}$ be a finite group scheme over an algebraically closed field $k$ of characteristic $\mathrm{char}(k)=p\ge 3$. In generalization of the familiar notion from the modular representation theory of finite groups, we define the $p$-rank ${\mathsf{rk}}_p(\mathcal{G})$ of $\mathcal{G}$ and determine the structure of those group schemes of $p$-rank $1$, whose linearly reductive radical is trivial. The most difficult case concerns infinitesimal groups of height $1$, which correspond to restricted Lie algebras. Our results show that group schemes of $p$-rank $\le 1$ are closely related to those being of finite or domestic representation type.

References in corpus (1)

• The elementary 3-Kronecker modules