Essential dimension of finite groups in prime characteristic

TL;DR

This paper determines the essential dimension of finite smooth group schemes over fields of prime characteristic, showing it is zero if the prime does not divide the group order and one if it does.

Contribution

It provides a complete characterization of the essential dimension of finite smooth group schemes in prime characteristic based on divisibility of the group order by the characteristic.

Findings

Essential dimension is 0 if prime does not divide group order.

Essential dimension is 1 if prime divides group order.

Abstract

Given a finite smooth group scheme $G$ over a field of characteristic $p > 0$, we show that the essential dimension of $G$ at $p$ is $0$ when $p$ does not divide the order of $G$, and $1$ when it does.