The solution of the \(k(GV)\)-problem.
type: Item
instance of: scholarly article
title: The solution of the \(k(GV)\)-problem.
zbMATH Open document ID: 1058.20010
author: David Gluck
author: Kay Magaard
author: Udo Riese
author: Peter Schmid
published in: Journal of Algebra
publication date: 2004-10-01T00:00:00Z
review text: Let \(H\) be a finite group and let \(p\) be a prime divisor of \(|H|\). Let \(B\) be a \(p\)-block of irreducible complex characters of \(H\) and let \(D\) be the defect group of \(B\). Let \(k(B)\) denote the number of irreducible characters of \(H\) in \(B\). Brauer's \(k(B)\)-conjecture is that \(k(B)\leq|D|\). In 1959, Brauer and Feit proved that \(k(B)\leq\tfrac 14|D|^2+1\) and this remains the best general bound obtained for \(k(B)\). While the \(k(B)\)-conjecture has remained fairly inaccessible, its verification for the family of \(p\)-solvable groups has seemed a reasonable goal, as one may hope to exploit the additional structure of such groups to force out a successful conclusion. The following is a typical problem faced when investigating such \(p\)-solvable groups. Let \(G\) be an arbitrary \(p'\)-group and let \(V\) be an elementary Abelian \(p\)-group on which \(G\) acts faithfully. Let \(GV\) denote the semi-direct product, where \(G\) acts naturally on the normal subgroup \(V\), and let \(k(GV)\) denote the number of conjugacy classes in \(GV\). The group \(GV\) is certainly \(p\)-solvable and elementary block theory shows that \(GV\) has only one \(p\)-block, namely the principal block \(B_1\), with defect group \(V\). Thus \(k(B_1)=k(GV)\) and, if correct, Brauer's conjecture implies that \(k(GV)\leq|V|\). Now in 1962, Nagao showed that, conversely, if \(k(GV)\leq|V|\) for all \(p'\)-groups \(G\) and all \(V\) on which \(G\) acts faithfully, then the \(k(B)\)-conjecture is true for all \(p\)-solvable groups. The \(k(GV)\)-problem is to prove the inequality \(k(GV)\leq|V|\). This paper is the conclusion of the proof that we do have \(k(GV)\leq|V|\), and thus the \(k(B)\)-conjecture is true for all \(p\)-solvable groups. It has taken just over 40 years since the appearance of Nagao's paper for the combined efforts of several mathematicians to be crowned by the final success of the four authors of this paper. It was not until the 1980's that any sub…
reviewed by: Rod Gow
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C15
zbMATH DE Number: 2104459
zbMATH Keywords: block theory
zbMATH Keywords: \(p\)-solvable groups
zbMATH Keywords: Brauer conjecture
zbMATH Keywords: real vectors
zbMATH Keywords: irreducible complex characters
zbMATH Keywords: defect groups
zbMATH Keywords: numbers of characters
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2004.02.027
OpenAlex ID: W4210741136
cites work: Q3684278
cites work: ON THE NUMBER OF IRREDUCIBLE CHARACTERS OF FINITE GROUPS IN A GIVEN BLOCK
cites work: Q4882944
cites work: Q3955560
cites work: On the k(GV) problem
cites work: The number of conjugacy classes in a finite group
cites work: The extraspecial case of the \(k(GV)\) problem
cites work: The \(k(GV)\) conjecture for modules in characteristic 31
cites work: Regular orbits of linear groups with an application to the \(k(GV)\)-problem. I
cites work: On the Number of Characters in a Block and the k (GV ) PROBLEM FOR SELF-DUAL V
cites work: Transitive linear groups and linear groups which contain irreducible subgroups of prime order. II
cites work: Q3952291
cites work: On the number of characters in a \(p\)-block of a \(p\)-solvable group
cites work: Regular orbits and the \(k(GV)\)-problem
cites work: On the number of conjugacy classes of a finite group
cites work: On the Orders of Maximal Subgroups of the Finite Classical Groups
cites work: Upper bounds for the number of conjugacy classes of a finite group
cites work: On the minimal degrees of projective representations of the finite Chevalley groups
cites work: Q5723973
cites work: The quasisimple case of the \(k(GV)\)-conjecture
cites work: On the extraspecial case of the \(k(GV)\)-conjecture
cites work: Counting conjugacy classes of certain semidirect products
cites work: Some remarks on the \(k(GV)\) problem
cites work: Further reductions for the \(k(GV)\)-problem
cites work: Self-dual modules and real vectors for solvable groups
cites work: Real vectors for linear groups and the \(k(GV)\)-problem
cites work: On Brauer's \(k(B)\)-problem
cites work: On Ree's Series of Simple Groups
DOI: 10.1016/J.JALGEBRA.2004.02.027
description: scientific article; zbMATH DE number 2104459