# On unipotent radicals of pseudo-reductive groups

## TL;DR

This paper investigates the structure of geometric unipotent radicals in pseudo-reductive groups over fields, providing bounds on their nilpotency class and analyzing element orders in specific cases.

## Contribution

It establishes bounds on the nilpotency class of unipotent radicals in pseudo-reductive groups and examines element orders in Weil restrictions over inseparable field extensions.

## Key findings

- Bounds on nilpotency class are sharp in many cases.
- Results on element orders in unipotent radicals of Weil restrictions.
- Structural insights into unipotent radicals of pseudo-reductive groups.

## Abstract

We establish some results on the structure of the geometric unipotent radicals of pseudo-reductive k-groups. In particular, our main theorem gives bounds on the nilpotency class of geometric unipotent radicals of standard pseudo-reductive groups, which are sharp in many cases. A major part of the proof rests upon consideration of the following situation: let k' be a purely inseparable field extension of k of degree p^e and let G denote the Weil restriction of scalars R_{k'/k}(G') of a reductive k'-group G'. When G= \R_{k'/k}(G') we also provide some results on the orders of elements of the unipotent radical \RR_u(G_{\bar k}) of the extension of scalars of G to the algebraic closure \bar k of k.