The Fitting length of the Hughes subgroup
The Fitting length of a solvable group is the least possible length of its normal series with nilpotent factors. Theorem: Let \(G\) be a finite solvable group having a normal subgroup \(H\neq G\) such that every element of \(G\setminus H\) is a \(p\)-element of order at most \(p^ n\), \(p\) odd. Then the Fitting length of \(H\) is at most \(n\). The bound is best possible. This theorem generalizes and strengthens earlier results of \textit{O. H. Kegel} [Math. Z. 75, 373-376 (1961; Zbl 0104.24904)] (the case \(n=1\)) and \textit{H. Kurzweil} [Arch. Math. 22, 136-145 (1971; Zbl 0218.20022)] who proved that Fitting length \(\leq 2n-2\) under the additional condition that \(p\) is not an exceptional prime in the Hall-Higman sense. The conditions of the theorem generalize the case of a fixed-point-free automorphism of order \(p^ n\) acting on a finite solvable \(p'\)-group. The Hall-Higman methods are used in the proof. The case \(p=2\) remains still open.
scientific article; zbMATH DE number 3975194