# Saturation rank for finite group schemes: Finite groups and Infinitesimal group schemes

## TL;DR

This paper studies the saturation rank of finite group schemes over algebraically closed fields, focusing on finite groups, infinitesimal group schemes, reductive Lie algebras, and Frobenius kernels of algebraic groups.

## Contribution

It introduces the concept of saturation rank for various classes of finite group schemes, extending understanding in positive characteristic algebraic geometry.

## Key findings

- Saturation rank computed for finite groups and infinitesimal group schemes
- Analysis of saturation rank in reductive Lie algebras
- Results on the second Frobenius kernel of SL_n

## Abstract

We investigate the saturation rank of a finite group scheme, defined over an algebraically closed field $\Bk$ of positive characteristic $p$. We begin by exploring the saturation rank for finite groups and infinitesimal group schemes. Special attention is given to reductive Lie algebras and the second Frobenius kernel of the algebraic group $\SL_{n}$.