Finite group schemes of essential dimension one

Abstract

We prove that if a finite group scheme $G$ over a field $k$ has essential dimension one, then it embeds in $PG{L}_{2/k}$. 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.

Corrected an error in the proof of the main theorem that was pointed out by D. Tossici. The results are unchanged