Embeddings of \(\text{PGL}_2(31)\) and \(\text{SL}_2(32)\) in \(E_8(\mathbb{C})\). With appendices by Michael Larsen and J. -P. Serre
type: Item
instance of: scholarly article
title: Embeddings of \(\text{PGL}_2(31)\) and \(\text{SL}_2(32)\) in \(E_8(\mathbb{C})\). With appendices by Michael Larsen and J. -P. Serre
zbMATH Open document ID: 0980.20035
DOI: 10.1215/S0012-7094-98-09409-1
author: Robert L. jun. Griess
author: Alex Ryba
published in: Duke Mathematical Journal
publication date: 2000-03-27T00:00:00Z
review text: The authors extend work begun by the first author and \textit{A.~M.~Cohen} [Proc. Symp. Pure Math. 47, Pt. 2, 367-405 (1987; Zbl 0654.22005)] to classify the finite simple subgroups of \(E_8(\mathbb{C})\). They announce a program to classify embeddings of \(\text{PGL}_2(31)\), \(\text{SL}_2(32)\), \(\text{PSL}_2(31)\), \(\text{PSL}_2(41)\), \(\text{PSL}_2(49)\) and \(Sz(8)\) into \(E_8(\mathbb{C})\), and begin in the current paper with the first two cases. The authors show that \(E_8(\mathbb{C})\) has three conjugacy classes of \(\text{PGL}_2(31)\)-subgroups, which yield three nonconjugate embeddings of the latter group into the former. Two of these were previously obtained by \textit{J.-P.~Serre} [Invent. Math. 124, No. 1-3, 525-562 (1996; Zbl 0877.20033)], who in an appendix to the present paper points out that the authors' results imply that \(\text{PGL}_2(31)\) embeds into \(E_8(p)\) for every prime \(p\). The authors also show that \(E_8(\mathbb{C})\) has a single conjugacy class of \(\text{SL}_2(32)\)-subgroups, and Serre's appendix notes that this implies \(E_8(q)\) contains \(\text{SL}_2(32)\) if the prime power \(q\) is conjugate to \(\pm 1\) modulo 11. The authors use a computational approach, beginning with consideration of a finite group \(L\) and prime \(p\) that does not divide the order of \(L\). They show that the number of conjugacy classes of \(L\)-subgroups in a quasisimple algebraic group in characteristic 0 is bounded by the number of conjugacy classes of \(L\)-subgroups in a corresponding algebraic group in characteristic \(p\). (An appendix by M.~Larsen uses algebraic-geometric methods to show in fact that equality actually holds in general.) The authors use their preliminary results to classify embeddings of \(\text{PGL}_2(31)\) and then \(\text{PSL}_2(32)\) into \(E_8(\mathbb{C})\) by studying embeddings into algebraic groups of type \(E_8\) in specially chosen prime characteristics: respectively, 2791 and 4093. They systematically describe …
Mathematics Subject Classification ID: 20G20
Mathematics Subject Classification ID: 20E32
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20D05
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20D08
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20G05
Mathematics Subject Classification ID: 20E45
zbMATH DE Number: 1425132
zbMATH Keywords: finite simple groups
zbMATH Keywords: embeddings
zbMATH Keywords: conjugacy classes
zbMATH Keywords: algebraic groups
zbMATH Keywords: liftings
zbMATH Keywords: quasisimple algebraic groups
zbMATH Keywords: finite simple subgroups
reviewed by: James F. Hurley
MaRDI profile type: Publication
cites work: Algebraic approximation of structures over complete local rings
cites work: Q5563439
cites work: Q3804918
cites work: Q3996420
cites work: Q3800308
cites work: The groupL(2 61) embeds in t h e Lie group of typeE8
cites work: On the Structure and Ideal Theory of Complete Local Rings
cites work: Q3684278
cites work: Elementary abelian p-subgroups of algebraic groups
cites work: Q4753716
cites work: On some Bruhat decomposition and the structure of the Hecke rings of \(p\)-adic Chevalley groups
cites work: Galois-Kohomologie halb einfacher algebraischer Gruppen über \(p\)-adischen Körpern. I, II
cites work: Q5588717
cites work: Maximality of Galois actions for compatible systems
cites work: Subfields and automorphism groups of \(p\)-adic fields
cites work: Q5752722
cites work: Q3871006
cites work: Unrefined minimal \(K\)-types for \(p\)-adic groups
cites work: Q4002684
cites work: First cohomology groups of some linear groups over fields of characteristic two
cites work: Q3031154
cites work: Q3929835
cites work: Q3856855
cites work: Examples for embeddings of groups \(\text{PSL}_ 2(F_ p)\) into simple Lie groups
cites work: Q4338613
cites work: Q3050630
cites work: Remarks on the cohomology of groups
cites work: Q3858229
Recommended article: Q3800308
Recommended article: Embeddings of \(\text{PSL}(2,41)\) and \(\text{PSL}(2,49)\) in \(E_8(\mathbb{C})\)
Recommended article: Classification of finite quasisimple groups which embed in exceptional algebraic groups
Recommended article: Finite simple groups which projectively embed in an exceptional Lie group are classified!
Recommended article: The groupL(2 61) embeds in t h e Lie group of typeE8
Recommended article: Conjugacy of 𝐴𝑙𝑡₅ and 𝑆𝐿(2,5) subgroups of 𝐸₈(𝐶)
Recommended article: Embeddings of \(U_ 3(8)\), \(Sz(8)\) and the Rudvalis group in algebraic groups of type \(E_ 7\)
Recommended article: Q4488486
Recommended article: Conjugacy of \(\text{Alt}_5\) and \(\text{SL}(2,5)\) subgroups of \(E_7(\mathbb{C})\)
Recommended article: GROUPS WITH EXACTLY TWO SUBGROUPS OF GIVEN ORDER
description: scientific article; zbMATH DE number 1425132