Finite groups with small conjugacy classes
type: Item
instance of: scholarly article
title: Finite groups with small conjugacy classes
zbMATH Open document ID: 0819.20024
DOI: 10.1007/BF02567696
author: Carlo Casolo
published in: Manuscripta Mathematica
publication date: 1995-08-24T00:00:00Z
full work available at URL: https://eudml.org/doc/155926
review text: Let \(G\) be a finite group. For \(g \in G\) let \(\sigma(g)\) denote the set of different primes dividing the class length \(| G: C_ G(g)|\). Put \(\sigma^*(G) = \max \{| \sigma(g)|\;g\in G\}\) and \(\rho^*(G) = \bigcup_{g \in G} \sigma(g)\). Huppert has conjectured that for solvable groups \(G\) always (I) \(| \rho^*(G)| \leq 2\sigma^* (G)\). Meanwhile this conjecture has attracted several authors and it has been verified for \(\sigma^*(G) \leq 2\). In a very surprising paper [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 2, 111-113 (1991; Zbl 0743.20018)] the author confirmed the inequality (I) for every perfect group. In the paper under review the author focuses on arbitrary groups with \(\sigma^*(G) = 2\) and solvable groups with \(\sigma^*(G) = 3\). It turns out that (I) holds true and equality occurs only for metabelian groups. Quite remarkable is the fact that the non-solvable groups with \(\sigma^*(G) = 2\) coincide exactly with those for which \(\sigma(G) = 1\) which have been classified by \textit{O. Manz} [J. Algebra 96, 114-119 (1985; Zbl 0567.20004)]. Here \(\sigma(G)\) is the maximum number of different primes dividing the degree of an irreducible character of \(G\). In analogy to the above, Huppert also conjectured that for solvable groups \(| \rho(G)| \leq 2\sigma(G)\) where \(\rho(G) = \{p \mid p\) prime, \(p \mid \chi(1)\) for some \(\chi \in \text{Irr}(G)\}\).
reviewed by: Wolfgang Willems
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20C15
zbMATH DE Number: 599657
zbMATH Keywords: lengths of conjugacy classes
zbMATH Keywords: character degrees
zbMATH Keywords: solvable groups
MaRDI profile type: Publication
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: Prime divisors of conjugacy class lengths in finite groups
cites work: On the length of the conjugacy classes of finite groups
cites work: Arithmetical conditions on the length of the conjugacy classes of a finite group
cites work: Connections between prime divisors of conjugacy classes and prime divisors of \(| G| \)
cites work: A conjecture about character degrees of solvable groups
cites work: Prime Factors of Character Degrees of Solvable Groups
cites work: Defect groups and character heights in blocks of solvable groups. II
cites work: Q3941595
cites work: Q5551230
cites work: Groups as Unions of Proper Subgroups
cites work: Endliche Gruppen I
cites work: Q3659962
cites work: Solvable groups character degrees and sets of primes
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cites work: Brauer characters of q'-degree in p-solvable groups
cites work: Q3834224
cites work: Q5507583
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description: scientific article; zbMATH DE number 599657