On Aschbacher's local \(C(G;T)\) theorem
type: Item
instance of: scholarly article
title: On Aschbacher's local \(C(G;T)\) theorem
zbMATH Open document ID: 0790.20029
DOI: 10.1007/BF02808113
author: Daniel Gorenstein
author: Richard Lyons
published in: Israel Journal of Mathematics
publication date: 1994-06-27T00:00:00Z
review text: This article is another important contribution of the authors to the revision of the classification of finite simple groups. We quote their abstract: Aschbacher's local \(C(G;T)\) theorem asserts that if \(G\) is a finite group with \(F^*(G) = O_ 2(G)\), and \(T \in \text{Syl}_ 2(G)\), then \(G = C(G;T)K(G)\), where \(C(G;T) = \langle N_ G(T_ 0)\mid 1 \neq T_ 0\text{ char }T\rangle\) and \(K(G)\) is the product of all near components of \(G\) of type \(L_ 2(2^ n)\) or \(A_{2^ n+1}\). Near components are also known as \(\chi\)-blocks or Aschbacher blocks. In this paper we give a proof of Aschbacher's theorem in the case that \(G\) is a \(K\)-group, i.e., in the case that every simple section of \(G\) is isomorphic to one of the known simple groups. Our proof relies on a result of \textit{U. Meierfrankenfeld} and \textit{G. Stroth} [Arch. Math. 55, No. 2, 105-110 (1990; Zbl 0719.20008)] on quadratic four-groups and on the Baumann- Glauberman-Niles theorem, for which \textit{B. Stellmacher} [Arch. Math. 46, 8-17 (1985; Zbl 0588.20013)] has given an amalgam-theoretic proof. Apart from those results, our proof is essentially self-contained.
Mathematics Subject Classification ID: 20D05
Mathematics Subject Classification ID: 20D40
zbMATH DE Number: 446711
zbMATH Keywords: revision
zbMATH Keywords: classification of finite simple groups
zbMATH Keywords: near components
zbMATH Keywords: \(\chi\)-blocks
zbMATH Keywords: Aschbacher blocks
zbMATH Keywords: \(K\)-group
zbMATH Keywords: quadratic four-groups
zbMATH Keywords: Baumann-Glauberman-Niles theorem
reviewed by: Ulrich Dempwolff
MaRDI profile type: Publication
cites work: On the Failure of the Thompson Factorization in 2-Constrained Groups
cites work: A Factorization Theorem for 2-Constrained Groups
cites work: GF(2)-Representations of Finite Groups
cites work: Involutions in Chevalley groups over fields of even order
cites work: Uber Endliche Gruppen Mit Einer ZU L 2 (2 n ) Isomorphen Faktorgruppe
cites work: Q3684278
cites work: FAILURE OF FACTORIZATION IN p-SOLVABLE GROUPS
cites work: Central elements in core-free groups
cites work: A Pair of Characteristic Subgroups for Pushing-Up in Finite Groups
cites work: Q3913574
cites work: The local structure of finite groups of characteristic 2 type
cites work: La structure des $p$-groupes de Sylow des groupes symétriques finis
cites work: Some subgroups of \(SI_ n(F_2)\)
cites work: On quadratic GF(2)-modules for Chevalley groups over fields of odd order
cites work: Pushing-up in finite groups
cites work: Einfache Gruppen, die von einer Konjugiertenklasse von Elementen der Ordnung drei erzeugt werden
cites work: Pushing up
cites work: Nonsolvable finite groups all of whose local subgroups are solvable
cites work: Groups with weakly closed TI-subgroups
description: scientific article; zbMATH DE number 446711