On Aschbacher’s local \(C(G;T)\) theorem. (English) Zbl 0790.20029
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 U. Meierfrankenfeld and 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 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.
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 U. Meierfrankenfeld and 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 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.
Reviewer: U.Dempwolff (Kaiserslautern)
MSC:
| 20D05 | Finite simple groups and their classification |
| 20D40 | Products of subgroups of abstract finite groups |
Keywords:
revision; classification of finite simple groups; near components; \(\chi\)-blocks; Aschbacher blocks; \(K\)-group; quadratic four-groups; Baumann-Glauberman-Niles theoremReferences:
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