Further reductions for the \(k(GV)\)-problem
type: Item
instance of: scholarly article
title: Further reductions for the \(k(GV)\)-problem
zbMATH Open document ID: 0882.20007
author: Geoffrey R. Robinson
published in: Journal of Algebra
publication date: 1998-03-09T00:00:00Z
review text: Let \(p\) be a prime, let \(G\) be a finite \(p'\)-group, and let \(V\) be a faithful (irreducible) \(\mathbb{F}_pG\)-module. Then \(k(GV)\) denotes the class number of the semidirect product \(GV\). The ``\(k(GV)\)-problem'' is to show that \(k(GV)\leq|V|\). A breakthrough on this problem was achieved in a joint paper of the author and \textit{J. G. Thompson} [J. Algebra 184, No. 3, 1143-1160 (1996)] where it was shown that the \(k(GV)\)-problem has a positive answer for primes \(p>5^{30}\). In the paper under review the methods of that paper are refined. In combination with results of D. Goodwin this yields a positive solution to the \(k(GV)\)-problem for all primes \(p>211\). As a consequence, for every prime \(p>211\), every \(p\)-block of every finite \(p\)-solvable group with defect group \(D\) contains at most \(|D|\) irreducible characters.
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20C10
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 1076269
zbMATH Keywords: Brauer's \(k(B)\)-problem
zbMATH Keywords: faithful modules
zbMATH Keywords: \(k(GV)\)-problem
zbMATH Keywords: \(p\)-blocks
zbMATH Keywords: finite \(p\)-solvable groups
zbMATH Keywords: irreducible modules
zbMATH Keywords: defect groups
zbMATH Keywords: irreducible characters
reviewed by: Burkhard Külshammer
MaRDI profile type: Publication
full work available at URL: jabr.1997.7085
OpenAlex ID: W1993296873
cites work: Regular orbits of linear groups
cites work: On Brauer's \(k(B)\)-problem
DOI: 10.1006/JABR.1997.7085
description: scientific article; zbMATH DE number 1076269