Fixed point free action with some regular orbits
type: Item
instance of: scholarly article
title: Fixed point free action with some regular orbits
zbMATH Open document ID: 0889.20015
author: Alexandre Turull
published in: Journal of Algebra
publication date: 1998-06-23T00:00:00Z
review text: Suppose that a finite group \(A\) acts on a finite (solvable) group \(G\) so that \(C_G(A)=1\). Assume further that \((|A|,|G|)=1\). There is the long-standing conjecture that the Fitting height of \(G\) is bounded by the length of the longest chain of subgroups of \(A\). \textit{A. Turull} [J. Reine Angew. Math. 371, 67-91 (1986; Zbl 0587.20017)]\ proved the conjecture in the case where for every proper subgroup \(B\) of \(A\) and every \(B\)-invariant elementary Abelian section \(S\) of \(G\), there exists some \(v\in S\) such that \(C_B(v)=C_B(S)\) (we say that \(B\) has regular orbit on \(S\)). In the present paper, the author generalizes this theorem by proving that the conclusion still holds when we assume that regular orbits exist for certain crucial sections only. The proof follows the outline of the proof in [loc. cit.].
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D30
zbMATH DE Number: 1066127
zbMATH Keywords: fixed point free actions
zbMATH Keywords: finite solvable groups
zbMATH Keywords: Fitting heights
zbMATH Keywords: chains of subgroups
zbMATH Keywords: elementary Abelian sections
zbMATH Keywords: regular orbits
reviewed by: N. Yu. Makarenko
MaRDI profile type: Publication
full work available at URL: jabr.1997.7035
OpenAlex ID: W2032834064
cites work: Supersolvable automorphism groups of solvable groups.
cites work: Fitting height of groups and of fixed points
cites work: Fixed point free action with regular orbits.
cites work: Generic fixed point free action of arbitrary finite groups
cites work: Cyclic by prime fixed point free action
DOI: 10.1006/JABR.1997.7035
description: scientific article; zbMATH DE number 1066127