On the coincidence of Gruenberg-Kegel graphs of an almost simple group and a nonsolvable Frobenius group
type: Item
instance of: scholarly article
title: On the coincidence of Gruenberg-Kegel graphs of an almost simple group and a nonsolvable Frobenius group
zbMATH Open document ID: 1506.20060
DOI: 10.1134/S0081543822030117
author: K. A. Ilenko
author: Natal'ya Vladimirovna Maslova
published in: Proceedings of the Steklov Institute of Mathematics
publication date: 2022-11-17T00:00:00Z
review text: Let \(G\) be a finite group and let \(\pi(G)\) be the set of prime divisors of \(|G|\). The Gruenberg-Kegel graph (or the prime graph) \(GK(G)\) is constructed as follows: the vertices of \(GK(G)\) are elements of the set \(\pi(G)\) and two vertices \(p,q\in\pi(G)\) are joined by an edge if and only if \(G\) contains an element of order \(pq\). This notion was introduced by Gruenberg and Kegel in the 1970s. One of the first results concerning prime graphs of groups was the unpublished Gruenberg-Kegel theorem, which described the structure of a finite group whose prime graph is disconnected. This theorem was published later in the paper by \textit{J. S. Williams} [J. Algebra 69, 487--513 (1981; Zbl 0471.20013)]. Frobenius groups and \(2\)-Frobenius groups play an important role in characterizing the structure of finite groups with the disconnected Gruenberg-Kegel graph. In this connection, in [Proc. Steklov Inst. Math. 283, S139--S145 (2013; Zbl 1307.20023); translation from Tr. Inst. Mat. Mekh. (Ekaterinburg) 18, No. 3, 99--105 (2012)] \textit{M. R. Zinov'eva} and \textit{V. D. Mazurov} stated the following question: which finite simple groups have the same Gruenberg-Kegel graph as Frobenius groups or \(2\)-Frobenius groups? In the same paper they found all finite simple groups having the same Gruenberg-Kegel graph as a Frobenius group or a \(2\)-Frobenius group. In the paper under review, the authors find all finite almost simple groups with the socle isomorphic to \(\mathrm{PSL}_2(q)\) for which there exists an nonsolvable Frobenius group with the same Gruenberg-Kegel graph. There are \(7\) such groups.
reviewed by: Mikhail Kabenyuk
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D05
Mathematics Subject Classification ID: 20D06
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 7618863
zbMATH Keywords: finite group
zbMATH Keywords: Gruenberg-Kegel graph
zbMATH Keywords: prime graph
zbMATH Keywords: Frobenius group
zbMATH Keywords: simple group
zbMATH Keywords: almost simple group
MaRDI profile type: Publication
full work available at URL: s0081543822030117
OpenAlex ID: W4312603538
cites work: Spectra of finite linear and unitary groups
cites work: Finite almost simple groups whose Gruenberg-Kegel graphs coincide with Gruenberg-Kegel graphs of solvable groups
cites work: Finite almost simple groups with prime graphs all of whose connected components are cliques
cites work: On finite groups with disconnected prime graph.
cites work: On finite tetraprimary groups.
cites work: Classification of maximal subgroups of odd index in finite simple classical groups.
cites work: On the coincidence of Grünberg-Kegel graphs of a finite simple group and its proper subgroup.
cites work: The maximal subgroups of the low-dimensional finite classical groups.
cites work: Q3684278
cites work: Q5577154
cites work: Q4375273
cites work: Decomposition of the Augmentation Ideal and of the Relation Modules of a Finite Group
cites work: Q4839675
cites work: Q3996618
cites work: Prime graph components of finite groups
cites work: Recognition of the simple groups L3(q) by element orders
Recommended article: Prime graph components of finite almost simple groups
Recommended article: On finite groups with disconnected prime graph.
Recommended article: The diameter of the prime graph of a finite group
Recommended article: On the connected components of the prime and Sylow graphs of a finite group
Recommended article: On some Frobenius groups with the same prime graph as the almost simple group PGL(2,49)
Recommended article: On the coincidence of Grünberg-Kegel graphs of a finite simple group and its proper subgroup.
Recommended article: On characterization by Gruenberg-Kegel graph of finite simple exceptional groups of Lie type
Recommended article: Criterion of unrecognizability of a finite group by its Gruenberg-Kegel graph
Recommended article: Addendum to ``Prime graph components of finite almost simple groups''.
Recommended article: Quasirecognition by the prime graph of the group \(C_n(2)\), where \(n\neq 3\) is odd.
description: scientific article; zbMATH DE number 7618863