## The Michigan Mathematical Journal

- Michigan Math. J.
- Volume 68, Issue 2 (2019), 277-299.

### On Unipotent Radicals of Pseudo-Reductive Groups

Michael Bate, Benjamin Martin, Gerhard Röhrle, and David I. Stewart

#### 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 $k\text{'}$ be a purely inseparable field extension of $k$ of degree ${p}^{e}$, and let $G$ denote the Weil restriction of scalars ${\mathrm{R}}_{k\text{'}/k}(G\text{'})$ of a reductive $k\text{'}$-group $G\text{'}$. When $G={\mathrm{R}}_{k\text{'}/k}(G\text{'})$, we also provide some results on the orders of elements of the unipotent radical ${\mathcal{R}}_{u}({G}_{\overline{k}})$ of the extension of scalars of $G$ to the algebraic closure $\overline{k}$ of $k$.

#### Article information

**Source**

Michigan Math. J., Volume 68, Issue 2 (2019), 277-299.

**Dates**

Received: 24 April 2017

Revised: 7 September 2018

First available in Project Euclid: 18 February 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.mmj/1550480563

**Digital Object Identifier**

doi:10.1307/mmj/1550480563

**Mathematical Reviews number (MathSciNet)**

MR3961217

**Zentralblatt MATH identifier**

07084763

**Subjects**

Primary: 20G15: Linear algebraic groups over arbitrary fields

#### Citation

Bate, Michael; Martin, Benjamin; Röhrle, Gerhard; Stewart, David I. On Unipotent Radicals of Pseudo-Reductive Groups. Michigan Math. J. 68 (2019), no. 2, 277--299. doi:10.1307/mmj/1550480563. https://projecteuclid.org/euclid.mmj/1550480563