On the length of the conjugacy classes of finite groups
type: Item
instance of: scholarly article
title: On the length of the conjugacy classes of finite groups
zbMATH Open document ID: 0694.20015
DOI: 10.1016/0021-8693(90)90168-N
author: David Chillag
author: Marcel Herzog
published in: Journal of Algebra
publication date: 1990-01-01T00:00:00Z
review text: If \(G\) is a finite group, then several results indicate that the structure of \(G\) is controlled to a large extent by arithmetical conditions on the degrees of the irreducible complex characters. In the paper under review the authors study the influence of the lengths of conjugacy classes of \(G\) on the structure of \(G\). There are surprising analogues to the character degree results. It turns out that in most cases arithmetical conditions on the class lengths restrict the structure of \(G\) more heavily than the same conditions imposed on the character degrees. For instance: A prime \(p\) does not divide \(| C|\) for each conjugacy class \(C\) of \(G\) if and only if the Sylow \(p\)-subgroup is contained in the center of \(G\). (This needs not the classification of finite simple groups while the analogue in the character degree case does (Ito-Michler Theorem).) To get the flavour of the many results the authors have the following may serve: Theorem 1: Suppose that \(| C|\) is squarefree for each conjugacy class \(C\) of \(G\). Then \(G\) is supersolvable and both \(| G/F(G)|\) and \(| F(G)'|\) are squarefree numbers. In particular, \(G'\leq F(G)\) and \(G/F(G)\) is cyclic. Moreover, the derived length of \(G\) is bounded by 3, the nilpotency class of \(F(G)\) by 2.
reviewed by: Wolfgang Willems
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20E34
Mathematics Subject Classification ID: 20D45
zbMATH DE Number: 4136210
zbMATH Keywords: irreducible complex characters
zbMATH Keywords: lengths of conjugacy classes
zbMATH Keywords: character degrees
zbMATH Keywords: Sylow subgroups
zbMATH Keywords: supersolvable groups
zbMATH Keywords: derived length
zbMATH Keywords: nilpotency class
MaRDI profile type: Publication
cites work: Q3804890
cites work: Q3695451
cites work: Group Elements of Prime Power Index
cites work: Q3684278
cites work: Group Characters and Normal Hall Subgroups
cites work: Q5577154
cites work: Endliche Gruppen I
cites work: Q3934684
cites work: Lineare auflösbare Gruppen
cites work: Degree-problems. I: Squarefree character degrees
cites work: Groups as Unions of Proper Subgroups
cites work: Q4103728
cites work: Solvable groups character degrees and sets of primes
cites work: A characterization of groups in terms of the degrees of their characters. II
cites work: Some Studies on Group Characters
cites work: On the Degrees of Irreducible Representations of a Finite Group
cites work: Q5675890
cites work: A note on character degrees of solvable groups
cites work: Q4771409
cites work: Q3803309
cites work: Endliche auflösbare Gruppen, deren sämtliche Charaktergrade Primzahlpotenzen sind. (Finite solvable groups all whose character degrees are prime powers)
cites work: Endliche nicht-auflösbare Gruppen, deren sämtliche Charaktergrade Primzahlpotenzen sind
cites work: Degree Problems: the p -Rank in p -Solvable Groups
cites work: A finite simple group of Lie type has p-blocks with different defects, \(p\neq 2\)
cites work: Groups with Normal, Solvable Hall p � -Subgroups
cites work: Q3941589
cites work: Finite Groups with Nilpotent Centralizers
cites work: Q3798851
cites work: Blocks of defect zero in finite simple groups of Lie type
description: scientific article; zbMATH DE number 4136210