A pushing up result and some consequences for the embedding of 2- constrained subgroups (Q803274)

The author considers the following situation. Let G be a finite group with \(Q_ 0=O_ 2(G)=F^*(G)\) and \(F^*(G/Q_ 0)\) a quasisimple group. Let S be a Sylow 2-subgroup of G, \(1\neq Z\leq \Omega_ 1(Z(S))\) and \(Q=O_ 2(N_ G(Z))\). Suppose that \(Q_ 0\leq Q_ 1\leq Q\), \(Q_ 1\trianglelefteq N_ G(Z)\) but no nontrivial characteristic subgroup of \(Q_ 1\) is normal in G. This is what is usually called a pushing-up situation. In this situation the author describes the quasisimple group \(F^*(G/Q_ 0)\) and its action on \(Q_ 0\cap O^ 2(G)\). There are too many possibilities to be stated here. But as a summary one gets that the quasisimple group is a symplectic, a linear group, or \(G_ 2(q)\). The modules involved in \(Q_ 0\cap O^ 2(G)\) are usually either trivial or natural modules (spin modules in case Sp(6,q)). The author gives some applications of his result. The applications are mainly to the following situation: Given a finite group G with subgroups \(M_ 0\) and \(M_ 1\) such that \(O_ 2(M_ 0)\subseteq O_ 2(M_ 1)\), for \(T\in Syl_ 2(M_ 0)\) there is some \(1\neq Z\leq \Omega_ 1(Z(T))\), \(Z\trianglelefteq M_ 1\). Under some further property on the structure of \(M_ 0\) (mainly generation properties) what can be said about \(M_ 0?\) These results are very interesting but too complicated to be stated here. The paper under review proves the power of the amalgam method. The author provides elegant and short proofs for rather complicated theorems.