The Fitting length of finite soluble groups. II: Fixed-point-free automorphisms
type: Item
instance of: scholarly article
title: The Fitting length of finite soluble groups. II: Fixed-point-free automorphisms
zbMATH Open document ID: 1401.20024
author: Enrico Jabara
published in: Journal of Algebra
publication date: 2017-07-27T00:00:00Z
review text: The aim of this paper is the obtain restrictions on the Fitting height of a finite soluble group \(G\) with a fixed-point-free automorphism \(\varphi\), that is, \(\text{C}_G(\varphi)=\{1\}\). We denote by \(h(G)\) the Fitting height of \(G\), by \(\pi(G)\) the set of primes dividing \(|G|\), by \(w(\varphi)\) the cardinality of \(\pi(\langle \varphi\rangle)\) and by \(W(\varphi)\) the composition length of \(\langle \varphi\rangle\) (that is, the number of prime divisors of \(\langle \varphi\rangle\) with multiplicities taken into account). It was conjectured that \(h(G)\leq W(\varphi)\), and it is known that if the order of \(\varphi\) is coprime to \(|G|\), then this conjecture holds. In this paper, the author considers the non-coprime case, in which the hypothesis that the order of \(G\) and \(\varphi\) are coprime is removed. The main result of this paper is Theorem~1.3: If \(w(\varphi)\geq 2\), then \(h(G)<(7w(\varphi)-9)W(\varphi)\leq 7W(\varphi)^2\). Some improvements of this result in some particular cases are obtained. Another result of this paper is Theorem~1.6: If \(G\) is a group and \(\varphi\) is an automorphism of \(G\), \(\gcd(|\text{C}_G(\varphi)|,|\langle\varphi\rangle|)=1\), \(w(\varphi)\geq 2\) and \(h(\text{C}_G(\varphi))=h_0\), then \(h(G)<(8w(\varphi)-10)W(\varphi)+\frac{3}{2}(w(\varphi)-1)w(\varphi)h_0\). For Part I see [\textit{G. Busetto} and the author, Arch. Math. 106, No. 5, 409--416 (2016; Zbl 1350.20015)].
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 6754375
zbMATH Keywords: fixed-point-free automorphism
zbMATH Keywords: Fitting length
zbMATH Keywords: soluble group
reviewed by: R. Esteban-Romero
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2017.06.002
OpenAlex ID: W2964343822
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DOI: 10.1016/J.JALGEBRA.2017.06.002
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description: scientific article; zbMATH DE number 6754375