Generalized fixed point free automorphisms
type: Item
instance of: scholarly article
title: Generalized fixed point free automorphisms
zbMATH Open document ID: 0605.20015
DOI: 10.1007/BF01191995
author: Alberto Espuelas
published in: Archiv der Mathematik
publication date: 1986-01-01T00:00:00Z
review text: In a previous paper [J. Algebra 105, 365-371 (1987; Zbl 0604.20021)] the author obtained the best possible bound for the Fitting length of a finite solvable group G with \(G\neq H_{p^ n}(G)\) where \(H_{p^ n}(G)=<x\in G|\) \(x^{p^ n}\neq 1>\) is the generalized Hughes subgroup. In the present article the author modifies his arguments from that paper to obtain a stronger result: Let G be a finite solvable group having a normal subgroup \(K\neq G\) such that every element in \(G\setminus K\) is a p-element, p an odd prime. If \(G\setminus K\) contains an element of order \(p^ n\), then the Fitting length of K is at most \(n+1\) and this bound is best possible. (In the previous paper \(''G\neq H_{p^ n}(G)''\) implies that \underbar{all} elements in \(G\setminus H_{p^ n}(G)\) are of order \(p^ n.)\) For \(p=2\) the analogue of the result above is false, as shown by means of an example.
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D25
Mathematics Subject Classification ID: 20D45
zbMATH DE Number: 3977239
zbMATH Keywords: Fitting length
zbMATH Keywords: finite solvable group
zbMATH Keywords: generalized Hughes subgroup
MaRDI profile type: Publication
cites work: The Fitting length of the Hughes subgroup
cites work: Q3913574
cites work: Finite p -Groups Acting on p -Soluble Groups
cites work: The \(H_p\)-problem and the structure of \(H_p\)-groups
cites work: Die Nilpotenz der \(H_ p\)-Gruppen
cites work: Eine Verallgemeinerung von fixpunktfreien Automorphismen endlicher Gruppen
cites work: The Fitting length of solvable \(H_{p^ n}\)-groups
description: scientific article; zbMATH DE number 3977239