On the diameter of a graph related to \(p\)-regular conjugacy classes of finite groups
type: Item
instance of: scholarly article
title: On the diameter of a graph related to \(p\)-regular conjugacy classes of finite groups
zbMATH Open document ID: 0964.20009
published in: Journal of Algebra
publication date: 2001-04-04T00:00:00Z
review text: Fix a prime \(p\) and a finite group \(G\). Consider the conjugacy classes of \(G\) for elements of order prime to \(p\), call this set \(\text{Con}_p(G)\). Now mimic a well-known construction of a graph by choosing as vertices the primes which divide the sizes of the classes in \(\text{Con}_p(G)\). Join two vertices if there is an element in \(\text{Con}_p(G)\) which is divisible by both primes. The authors prove a number of theorems about this graph for \(p\)-soluble groups. For example that the number of connected components is at most 2 (Theorem 1). If \(G\) is soluble with two connected components then the primes dividing \(|G|\) other than \(p\) can be divided into two sets, \(\pi_1\) and \(\pi_2\) so that at least one Hall \(\pi_i\)-subgroup is metabelian.
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 05C25
Mathematics Subject Classification ID: 20D20
Mathematics Subject Classification ID: 20E45
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 1526433
zbMATH Keywords: conjugacy classes
zbMATH Keywords: finite groups
zbMATH Keywords: graphs
zbMATH Keywords: \(p\)-soluble groups
zbMATH Keywords: connected components
zbMATH Keywords: Hall subgroups
author: Ziqun Lu
author: Jiping Zhang
reviewed by: A. R. Camina
MaRDI profile type: Publication
full work available at URL: jabr.2000.8354
OpenAlex ID: W2027121942
cites work: On a Graph Related to Conjugacy Classes of Groups
cites work: Applications of a graph related to conjugacy classes in finite groups
cites work: On the length of the conjugacy classes of finite groups
cites work: On the Diameter of a Graph Related to Conjugacy Classes of Groups
cites work: Arithmetical conditions on the length of the conjugacy classes of a finite group
cites work: Prime Factors of Conjugacy Classes of Finite Solvable Groups
cites work: The conjugacy classes in a finite simple group.
cites work: Endliche Gruppen I
cites work: Q4273959
DOI: 10.1006/JABR.2000.8354
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description: scientific article; zbMATH DE number 1526433