Simultaneous pushing up
The pushing up problem for a finite group P is the following: If G is a p-constrained group with \(G/O_ p(G)\cong P\) and \(S\in Syl_ p(G)\), then either find a characteristic subgroup of S normal in G, or describe the structure of G. In the paper under review the author handles a certain generalization of this problem. Let G be a group generated by a collection \(\{P_ i\}_{i\in I}\) of subgroups such that \(O^{p'}(P_ i/O_ p(P_ i))\) is a perfect central extension of a rank 1 Lie-type group in characteristic p. Furthermore there is \(S\leq \cap_{i\in I}P_ i\), \(S\in Syl_ p(P_ i)\) for all i. Finally assume that \(Z=\Omega_ 1(Z(S))\) is not normal in any \(P_ i\). To have relations between the \(P_ i's\) the author introduces a graph \({\tilde \Gamma}\)(I) where two vertices \(i,j\in I\) are connected if \(O_ p(P_ i)\cap O_ p(P_ j)\ntrianglelefteq O^{p'}(P_ i)\). Then \({\tilde \Gamma}\)(I) becomes a directed graph. Now under certain assumptions on \({\tilde \Gamma}\)(I) the author proves a very general pushing-up result, too technical to be stated here. But to see the type of results obtained one corollary should be stated. Under the assumption that \({\tilde \Gamma}\)(I) is connected (as directed graph) he proves that one of the following holds (i) \(J(S)\triangleleft G\), (ii) \([\Phi (S),S]\triangleleft G\), or (iii) \(p=3\) and \(K_ 3(S)\triangleleft G\). The proof of the results uses heavily methods which are known as the amalgam method.
scientific article; zbMATH DE number 4089789