Finite group schemes of $p$-rank $\leq1$

TL;DR

This paper classifies finite group schemes over algebraically closed fields with characteristic p≥3 that have p-rank at most 1, revealing their structure and relation to representation types.

Contribution

It introduces the p-rank for finite group schemes and characterizes those with p-rank 1, especially infinitesimal groups of height 1, linking them to finite or domestic representation types.

Findings

Classified group schemes of p-rank ≤ 1 with trivial linearly reductive radical.

Connected p-rank 1 infinitesimal groups correspond to restricted Lie algebras.

Established relation between low p-rank group schemes and finite or domestic representation types.

Abstract

Let $\mathcal{G}$ be a finite group scheme over an algebraically closed field $k$ of characteristic ${\rm char}(k)=p\geq 3$. 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 $\leq 1$ are closely related to those being of finite or domestic representation type.