# Finite group schemes of essential dimension one

## TL;DR

This paper characterizes finite group schemes of essential dimension one, showing they embed in PGL_2 and providing explicit classifications over various fields.

## Contribution

It proves embedding properties for finite group schemes of essential dimension one and classifies infinitesimal schemes over any field.

## Key findings

- Finite group schemes of essential dimension one embed in PGL_2.
- Explicit classification of infinitesimal group schemes of essential dimension one.
- Characterization of such schemes over algebraically closed fields.

## Abstract

We prove that if a finite group scheme $G$ over a field $k$ has essential dimension one, then it embeds in $PGL_{2/k}$. We use this to give an explicit classification of all infinitesimal group schemes of essential dimension one over any field and a characterisation of all finite group schemes of essential dimension one over algebraically closed fields.