A Fitting height lemma and its applications
type: Item
instance of: scholarly article
title: A Fitting height lemma and its applications
zbMATH Open document ID: 1478.20010
author: M. Yasir Kızmaz
published in: Archiv der Mathematik
publication date: 2021-07-13T00:00:00Z
full work available at URL: http://hdl.handle.net/11693/76910
review text: Let \(A\) be a group acting on a finite solvable group \(G\) and let \(N\) be an \(A\)-invariant normal subgroup of \(G\) such that \([G,A]\not\subseteq N.\) The author proves the inequality \(h([G,A])\leq h([G,A]N/N) + h([N,A])\) where \(h(G)\) denotes the Fitting height of \(G.\) As an application of this result, he obtains several Fitting height inequalities, involving the new concept of ``fixed point free separability''. An \(A\)-invariant subnormal series of \(G,\) \(1=N_k \unlhd \cdots \unlhd N_0=G,\) is called a fixed point free series of \(G\) with respect to \(A\) if the action of \(A\) on each quotient \(N_i/N_{i+1}\) is either trivial or fixed point free. The group \(G\) is said to be fixed point free separable if there is a fixed point free series of \(G\) with respect to each subgroup \(A\) of Aut\((G)\). A new characterization of solvable groups is given: a finite group \(G\) is solvable if and only if it is fixed point free separable.
reviewed by: Andrea Lucchini
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 7371066
zbMATH Keywords: Fitting height
zbMATH Keywords: solvable groups
zbMATH Keywords: fixed point free separability
MaRDI profile type: Publication
OpenAlex ID: W3157992464
Wikidata QID: Q124881520
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cites work: The Fitting length of finite soluble groups. II: Fixed-point-free automorphisms
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DOI: 10.1007/S00013-021-01591-3
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description: scientific article; zbMATH DE number 7371066