On Aschbacher's local \(C(G;T)\) theorem (Q689853)

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.