Aspects of the conjugacy class structure of simple algebraic groups.
type: Item
instance of: scholarly article
title: Aspects of the conjugacy class structure of simple algebraic groups.
zbMATH Open document ID: 1214.20044
author: Martin Cook
published in: Journal of Algebraic Combinatorics
publication date: 2010-05-05T00:00:00Z
review text: Let \(G\) be an adjoint simple algebraic group over an algebraically closed field of characteristic \(p\), \(\Phi\) the root system of \(G\), and \(t\in\mathbb{N}\). Lawther has proven that the dimension of the set \(G_{[t]}=\{g\in G:g^t=1\}\) depends only on \(\Phi\) and \(t\). In particular the dimension is independent of \(p\); which was observed for small \(t\) by Liebeck. Since \(G_{[t]}\) is clearly a disjoint union of conjugacy classes, the question arises whether a similar result holds if we replace \(G_{[t]}\) by one of those classes. This paper provides a partial answer to that question. E.g., take \(p,q\) to be distinct primes and \(G_p\) and \(G_q\) to be adjoint simple algebraic groups with the same root system and over algebraically closed fields of characteristic \(p\) and \(q\), respectively. If \(s\in G_p\) has order \(q\) then there exists an element \(u\in G_q\) of the same order such that \(\dim u^{G_q}=\dim s^{G_p}\).
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20E45
zbMATH DE Number: 5703814
zbMATH Keywords: simple algebraic groups
zbMATH Keywords: root systems
zbMATH Keywords: conjugacy classes
zbMATH Keywords: semisimple elements
zbMATH Keywords: unipotent elements
zbMATH Keywords: characteristic independent
reviewed by: L. N. Vaserstein
MaRDI profile type: Publication
full work available at URL: s10801-009-0187-y
OpenAlex ID: W2053103143
cites work: Q3682663
cites work: Centralizers of Elements in Locally Finite Simple Groups
cites work: Nilpotency in classical groups over a field of characteristic 2
cites work: Jordan block sizes of unipotent elements in exceptional algebraic groups
cites work: Elements of specified order in simple algebraic groups
DOI: 10.1007/S10801-009-0187-Y
description: scientific article; zbMATH DE number 5703814