Fixed point free action on groups of odd order.
type: Item
instance of: scholarly article
title: Fixed point free action on groups of odd order.
zbMATH Open document ID: 1155.20021
author: Gülin Ercan
author: İsmail Ş. Güloğlu
published in: Journal of Algebra
publication date: 2008-10-06T00:00:00Z
full work available at URL: https://hdl.handle.net/11511/28624
review text: Let \(G\) be a finite solvable group, let \(A\) be a nilpotent group of automorphisms of \(G\), and assume that \(C_G(A)=1\). It is conjectured that then the Fitting height \(h(G)\) of \(G\) is bounded above by the length \(\ell(A)\) of the longest chain of subgroups of \(A\). The authors prove this conjecture when \(G\) has odd order and \(A\) is Abelian of square-free exponent prime to \(6\). This improves, in this case, the linear bound proved by \textit{A. Turull} [in NATO ASI Ser., Ser. C, Math. Phys. Sci. 471, 377-400 (1995; Zbl 0847.20002)] to the best possible bound. Both the authors' and the reviewer's proofs follow the outline of the proof by \textit{E. C. Dade} [Ill. J. Math. 13, 449-514 (1969; Zbl 0195.04003)] together with the observation that for an Abelian group \(A\) of square free exponent acting fixed-point freely on a Fitting chain, one will be able to find a normal subgroup of \(A\) of prime order acting non-trivially on the top of the Fitting chain. More detailed analysis in the present paper yields the authors' stronger result.
reviewed by: Alexandre Turull
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 5349061
zbMATH Keywords: finite solvable groups
zbMATH Keywords: Fitting heights
zbMATH Keywords: fixed point free actions
zbMATH Keywords: representations
MaRDI profile type: Publication
OpenAlex ID: W2044539321
cites work: A NOTE ON FIXED-POINT-FREE ACTIONS OF FINITE GROUPS
cites work: Carter subgroups and Fitting heights of finite solvable groups
cites work: Finite groups admitting fixed-point free automorphisms of order pqr
cites work: Finite groups admitting fixed point free automorphisms of order pq
cites work: Fitting height of groups and of fixed points
cites work: Q4856489
DOI: 10.1016/J.JALGEBRA.2008.01.033
description: scientific article; zbMATH DE number 5349061