The normal hull and commutator group for nonconnected group schemes

TL;DR

This paper establishes a well-defined notion of normal hull and commutator group for smooth algebraic group schemes over a field, extending concepts to nonconnected cases.

Contribution

It introduces a consistent definition of normal hull and commutator group for smooth algebraic group schemes, including nonconnected cases.

Findings

Normal hull is well-behaved for smooth algebraic group schemes.

Commutator group is well-defined for nonconnected smooth group schemes.

Extends classical notions to broader class of algebraic groups.

Abstract

In this short note, we prove that there is a well behaved notion of normal hull for smooth algebraic group schemes over a field and that the commutator group $(G,H)$ is well defined for $H\subset G$ smooth, even when both of them are not connected.