Relation between properties of intersections of \(S_ \pi\)-subgroups and \(\pi\)-length in a finite \(\pi\)-solvable group (Q1311102)

Let \(G\) be a finite group of order \(|G|\). An \(S_\pi\)-subgroup of \(G\) is a subgroup of order \(|G|_\pi\), where \(|G|_\pi\) is the greatest \(\pi\)-divisor of \(|G|\). Bounds for the \(\pi\)-length of a finite \(\pi\)-solvable group in which the intersections of an \(S_\pi\)-subgroup \(H\) with all its conjugates in the group satisfy a certain condition for \(x\in G\setminus N_G(H)\) are derived. In particular the theorem of Hall and Higman, according to which the \(p\)-length of a finite \(p\)-solvable group cannot exceed the nilpotency class of an \(S_p\)-subgroup, is a corollary of the results given.