Finite simple groups of Lie type have non-principal p-blocks, p\(\neq 2\)
type: Item
instance of: scholarly article
title: Finite simple groups of Lie type have non-principal p-blocks, p\(\neq 2\)
zbMATH Open document ID: 0608.20004
DOI: 10.1016/0021-8693(85)90206-6
author: Peter Brockhaus
author: Gerhard O. Michler
published in: Journal of Algebra
publication date: 1985-01-01T00:00:00Z
review text: The paper gives a proof of the titular theorem. After the introduction, section 2 gives the prerequisites from block theory. Section 3 summarizes known results on maximal tori and the structure of semisimple elements of finite groups G of Lie type. In section 4, the proof proceeds by destroying the minimal counterexample L; here \(L\cong G/Z(G)\). By case by case analysis, it is shown that every p-element x in \(G\setminus Z(G)\) is regular, using the centralizer \(C_ G(x)\) of semisimple x; further \(p\nmid | Z(G)|\). The argument concludes by using a theorem of Boyce to show that a Sylow p-subgroup S of G must be cyclic and in this case a theorem of the second author implies that L must have at least two blocks, the final contradiction.
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20G40
Mathematics Subject Classification ID: 20D06
zbMATH DE Number: 3983455
zbMATH Keywords: finite groups of Lie type
zbMATH Keywords: non-principal p-block
zbMATH Keywords: semisimple elements
zbMATH Keywords: Sylow p-subgroup
zbMATH Keywords: blocks
MaRDI profile type: Publication
full work available at URL: 0021-8693(85)90206-6
OpenAlex ID: W2053481099
Wikidata QID: Q115366903
cites work: Involutions in Chevalley groups over fields of even order
cites work: Irreducible representations of finite groups of Lie type through block theory and special conjugacy classes
cites work: Some applications of the theory of blocks of characters of finite groups. I
cites work: On the radical of a group algebra
cites work: Über die Zerlegungszahlen der Suzukigruppen Sz(q)
cites work: Q5668957
cites work: Centralizers of Semisimple Elements in the Finite Classical Groups
cites work: Q4073555
cites work: The centralizers of semisimple elements of the Chevalley groups \(E_ 7\) and \(E_ 8\)
cites work: Q3693709
cites work: Q5645586
cites work: Q3955560
cites work: On finite simple groups of characteristic 2 type
cites work: Endliche Gruppen I
cites work: Q4103728
cites work: Non-solvable finite groups with cyclic Sylow p-subgroups have non- principal p-blocks
cites work: Q4183492
cites work: Q4063618
cites work: Q4405090
cites work: The Character Tables for SL(3, q), SU(3, q2), PSL(3, q), PSU(3, q2)
cites work: Q5662587
cites work: Representations of Algebraic Groups
cites work: Endomorphisms of linear algebraic groups
cites work: Q4149898
cites work: On Ree's Series of Simple Groups
description: scientific article; zbMATH DE number 3983455