On the surjectivity of the power maps of semisimple algebraic groups.
type: Item
instance of: scholarly article
title: On the surjectivity of the power maps of semisimple algebraic groups.
zbMATH Open document ID: 1045.20043
author: Pralay Chatterjee
published in: Mathematical Research Letters
publication date: 2004-05-18T00:00:00Z
review text: The author formulates and proves the analogue in positive characteristic of theorem C of \textit{P. Chatterjee} [Math. Res. Lett. 9, No. 5-6, 741-756 (2002; Zbl 1025.20034)]. This had been done a little earlier by \textit{R. Steinberg} [Math. Res. Lett. 10, No. 5-6, 621-624 (2003; see the following review Zbl 1045.20044)], but the proofs differ. Next he applies the result to tell when a connected semisimple algebraic group is `exponential' in the sense that every element is contained in a connected Abelian algebraic subgroup. Say the characteristic of the ground field is \(p>0\). Then the following are equivalent: 1.~\(G\) is exponential. 2.~The map \(x\mapsto x^n\) is surjective for all positive integers \(n\) prime to \(p\) and \(p\) is a good prime for \(G\). 3.~\(G\) is isogenous to a product of groups of type \(A_\ell\), \(\ell\geq 1\), and the reduced center of \(G\) is trivial.
reviewed by: Wilberd van der Kallen
Mathematics Subject Classification ID: 20G15
zbMATH DE Number: 2064719
zbMATH Keywords: bad primes
zbMATH Keywords: power maps
zbMATH Keywords: exponential algebraic groups
MaRDI profile type: Publication
full work available at URL: mrl.2003.v10.n5.a6
OpenAlex ID: W1967998025
DOI: 10.4310/MRL.2003.V10.N5.A6
description: scientific article; zbMATH DE number 2064719