Finite groups admitting a \(p\)-solvable operator group
Consider a \(\pi\)-solvable operator group \(H\) acting on a group \(G\). The author proves that if \(G\) has a nilpotent maximal \(H\)-invariant subgroup which has a Sylow 2-subgroup with no section isomorphic to the dihedral group of order eight, then \(G\) is solvable. This generalizes results on nilpotent maximal subgroups by J. G. Thompson and Z. Janko. Also a group \(G\) is solvable if \(H\) acts nontrivially on \(G\) but trivially on each proper \(H\)-invariant subgroup, in fact \(G\) is metabelian with a well-defined structure. However if \(H\) acts trivially on each \(H\)-invariant metabelian subgroup of \(G\), then \(H\) acts trivially on \(G\). Several other similar results are developed and examples are provided to highlight the necessity for the constraints in the hypotheses.
nilpotent maximal \(H\)-invariant subgroups
scientific article; zbMATH DE number 1218686