Examples for embeddings of groups \(\text{PSL}_ 2(F_ p)\) into simple Lie groups
type: Item
instance of: scholarly article
title: Examples for embeddings of groups \(\text{PSL}_ 2(F_ p)\) into simple Lie groups
zbMATH Open document ID: 0877.20033
published in: Inventiones Mathematicae
publication date: 1997-10-22T00:00:00Z
review text: The main theorem is as follows: Let \(G\) be a connected simple linear algebraic group over an algebraically closed field \(k\) with Coxeter number \(h\). Then (1) If \(p=h+1\), then \(G(k)\) contains a subgroup isomorphic to \(\text{PGL}_2(\mathbb{F}_p)\), except when \(\text{char}(k)=2\), \(G\cong\text{PGL}_2\); (2) If \(p=2h+1\), then \(G(k)\) contains a subgroup isomorphic to \(\text{PSL}_2(\mathbb{F}_p)\). To prove this theorem, first under the assumption that \(\text{char}(k)=p\geq h\) the author constructs a ``principal'' embedding \(\text{PGL}_2(\mathbb{F}_p)\to G(\mathbb{F}_p)\). If \(p=h+1\), then the above embedding can be lifted to an embedding \(\text{PGL}_2(\mathbb{F}_p)\to G(\mathbb{Z}_p)\) where \(\mathbb{Z}_p\) is the ring of \(p\)-adic integers. If \(p=2h+1\), then it can be lifted to an embedding \(\text{PSL}_2(\mathbb{F}_p)\to G(\mathbb{Z}_p[\sqrt{\pm p}])\). Finally the Bruhat-Tits theory is used to complete the base change from characteristic zero to arbitrary characteristic.
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E32
zbMATH DE Number: 866101
zbMATH Keywords: simple Lie groups
zbMATH Keywords: finite subgroups
zbMATH Keywords: connected simple linear algebraic groups
zbMATH Keywords: Coxeter numbers
zbMATH Keywords: embeddings
zbMATH Keywords: base changes
MaRDI profile type: Publication
full work available at URL: s002220050062
OpenAlex ID: W2007529579
DOI: 10.1007/S002220050062
author: Jean-Pierre Serre
description: scientific article; zbMATH DE number 866101