On the connected components of the prime and Sylow graphs of a finite group
type: Item
instance of: scholarly article
title: On the connected components of the prime and Sylow graphs of a finite group
zbMATH Open document ID: 1497.20025
published in: Archiv der Mathematik
publication date: 2022-03-14T00:00:00Z
review text: Let \(G\) be a finite group and let \(\pi(G)\) be the set of prime divisors of \(|G|\). The prime graph (or Gruenberg-Kegel graph) \(\Gamma_p(G)\) is constructed as follows: the vertices of \(\Gamma_p(G)\) 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 One of the first results concerning prime graphs of groups was the unpublished Gruenberg-Kegel theorem, which described the structure of a finite group whose prime graph is disconnected.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 a paper by \textit{J. S. Williams} [J. Algebra 69, 487--513 (1981; Zbl 0471.20013)] In the following years, many different results about the structure of a prime graph of a finite group were obtained. In particular, the diameter of this graph is at most \(5\) [\textit{M. S. Lucido}, J. Group Theory 2, No. 2, 157--172 (1999; Zbl 0921.20020)]. The Sylow graph \(\Gamma_s(G)\) of a group \(G\) is a graph with \(V(\Gamma_s(G)) = \pi(G)\) and two vertices \(p\), \(q\) in form an edge of if either \(q\in\pi(N_G(G_p)/G_pC_G(G_p))\) or \(p\in\pi(N_G(G_q)/G_qC_G(G_q))\), where \(G_p\) is a Sylow \(p\)-subgroup of \(G\). The Sylow graph has been studied by \textit{L. S. Kazarin} et al. in [Isr. J. Math. 186, 251--271 (2011; Zbl 1280.20027)]. They proved that if \(G\) is an almost simple group, then the graph \(\Gamma_s(G)\) is connected and has diameter at most \(5\). The connectivity of \(\Gamma_s(G)\) was proved involving the classification of finite simple groups. \textit{F. G. Russo} in [``Problems of connectivity between the Sylow graph, the prime graph and the non-commuting graph of a group'', Adv. Pure Math. 2, No. 6, 391--396 (2012, \url{doi:10.4236/apm.2012.26058 })] pointed out that if \(G\) is an almost simple finite group, then diameters of graphs \(\Gamma_p(G)\) and \(\Gamma_s(G)\) are at most \(5\). In this connection, he formulated …
reviewed by: Mikhail Kabenyuk
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D05
Mathematics Subject Classification ID: 20D06
zbMATH DE Number: 7489065
zbMATH Keywords: finite group
zbMATH Keywords: connected graph
zbMATH Keywords: prime graph
zbMATH Keywords: Sylow graph
author: V. I. Murashko
MaRDI profile type: Publication
full work available at URL: s00013-021-01694-x
OpenAlex ID: W4205241133
Wikidata QID: Q114232133
cites work: Non-commuting graph of a group.
cites work: A generalization of prime graphs of finite groups
cites work: On normalizers of Sylow subgroups in finite groups
cites work: Classes of Finite Groups
cites work: Q3567580
cites work: Finite soluble groups
cites work: The influence of the complete nonexterior square graph on some infinite groups
cites work: Prime-power factor groups of finite groups
cites work: Prime-power factor groups of finite groups. II
cites work: On the class of Sylow tower groups
cites work: The Sylow structure of scalar automorphism groups
cites work: Endliche Gruppen I
cites work: On the Sylow graph of a group and Sylow normalizers.
cites work: PRIME GRAPH COMPONENTS OF FINITE SIMPLE GROUPS
cites work: The diameter of the prime graph of a finite group
cites work: Non-permutability graph of subgroups
cites work: Non-exterior square graph of finite group
cites work: Permutability graphs of subgroups of some finite non-abelian groups
cites work: Arithmetic graphs and classes of finite groups
cites work: Prime graph components of finite groups
DOI: 10.1007/S00013-021-01694-X
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description: scientific article; zbMATH DE number 7489065