Irreducible disconnected systems in groups
type: Item
instance of: scholarly article
title: Irreducible disconnected systems in groups
zbMATH Open document ID: 0936.20020
DOI: 10.1007/BF02810685
author: Gil Kaplan
published in: Israel Journal of Mathematics
publication date: 2000-02-27T00:00:00Z
review text: Let \(G\) be an arbitrary group with subgroup \(A\). The index \([A:A\cap A^g]\) (\(g\in G\)) is called a subdegree of the pair \((A,G)\). This concept was presented by \textit{I. M. Isaacs} and \textit{C. E. Praeger} [J. Algebra 159, No. 1, 158-175 (1993; Zbl 0815.20002)]. Let \(D\) denote the set of all subdegrees of \((A,G)\). Assume further that \(A\) is not normal in \(G\) and all the subdegrees of \((A,G)\) are finite. Then the common divisor graph \(\Gamma\) is associated to \((A,G)\): its vertex set is \(D-\{1\}\), and two different subdegrees \(s,t\) are joined by an edge if \(\gcd(s,t)\not=1\). In the above mentioned paper, it is proved that \(\Gamma\) has at most two connected components. Assume that \(\Gamma\) is disconnected. Let \(D_1\) be the component of \(\Gamma\) containing \(\min(D-\{1\})\). In [J. Algebra 193, No. 2, 616-628 (1997; Zbl 0891.20018)], the author proved that if \(A\) is stable in \(G\) (a property which holds when \(A\) or \([G:A]\) is finite), then the set \(H=\{g\in G\mid[A:A\cap A^g]\in D_1\}\cup\{1\}\) is a subgroup of \(G\). In this case he says that \(A<H<G\) is a disconnected system (briefly: a system) of \(G\). In this paper, the author deals with some fundamental types of systems. A system \(A<H<G\) is irreducible if there does not exist a system of the form \(AN/N<H/N<G/N\), where \(1<N\triangleleft G\), \(AN<H\). Theorem A gives restrictions on the finite nilpotent normal subgroups of \(G\), when \(G\) possesses an irreducible system. In particular, if \(G\) is finite then the Fitting subgroup \(\text{Fit}(G)\) of \(G\) is primary. The author considers also general systems. Corollary 4.2. says that if \(G\) is finite and \(G\) possesses a system \(A<H<G\) then \(G\) has a proper normal subgroup \(N\) such that \(AN/N\) is not nilpotent and \(\text{Fit}(G/N)\) is primary. Theorem B says that for any system \(A<H<G\), \(N_G(N_G(A))=N_G(A)\). Theorem C and Corollary C' generalize Theorem 2 from a paper of \textit{C. E. Praeg…
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20E15
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20D25
Mathematics Subject Classification ID: 20D30
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 1356477
zbMATH Keywords: groups
zbMATH Keywords: subdegrees
zbMATH Keywords: common divisor graphs
zbMATH Keywords: irreducible systems in groups
zbMATH Keywords: finite nilpotent normal subgroups
zbMATH Keywords: Fitting subgroups
reviewed by: Anatoli Kondrat'ev
MaRDI profile type: Publication
cites work: Subgroups close to normal subgroups
cites work: Permutation group subdegrees and the common divisor graph
cites work: On groups admitting a disconnected common divisor graph
cites work: Q5512231
description: scientific article; zbMATH DE number 1356477