On the number of conjugacy classes of a finite solvable group. II.
type: Item
instance of: scholarly article
title: On the number of conjugacy classes of a finite solvable group. II.
zbMATH Open document ID: 1047.20016
author: Lászlo Héthelyi
author: Burkhard Külshammer
published in: Journal of Algebra
publication date: 2004-02-03T00:00:00Z
review text: [For part I see Bull. Lond. Math. Soc. 32, No. 6, 668-672 (2000; Zbl 1024.20016).] This paper proves the following theorem: Let \(G\) be a finite soluble group and let \(p\) be a prime where \(p^2\) divides the order of \(G\). Then \(G\) has at least \((49p+1)/60\) conjugacy classes. Since we know that the number of conjugacy classes is the same as the number of irreducible characters, the same result applies to the number of irreducible characters. The authors use this fact and really prove it in that form. A key result is the observation that the Alperin-McKay conjecture holds for soluble groups [\textit{T. Okuyama} and \textit{M. Wajima}, Osaka J. Math. 17, 801-806 (1980; Zbl 0446.20003)]. This enables the authors to use induction and show that they need only consider the case where the Sylow \(p\)-subgroup is a normal elementary Abelian subgroup of \(G\) and \(O_{p'}(G)=1\).
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20E45
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 2037081
zbMATH Keywords: finite soluble groups
zbMATH Keywords: numbers of conjugacy classes
reviewed by: A. R. Camina
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2003.05.002
OpenAlex ID: W2088757328
cites work: Q5634146
cites work: Solvable Primitive Permutation Groups of Low Rank
cites work: On The Number of Conjugacy Classes of a Finite Solvable Group
cites work: Endliche Gruppen I
cites work: Q3659962
cites work: Q4273959
cites work: Character correspondence and p-blocks of P-solvable groups
cites work: The rank of a finite primitive solvable permutation group
cites work: Classification of finite groups according to the number of conjugacy classes. II
DOI: 10.1016/J.JALGEBRA.2003.05.002
description: scientific article; zbMATH DE number 2037081