On the k(GV) problem
type: Item
instance of: scholarly article
title: On the k(GV) problem
zbMATH Open document ID: 0536.20008
DOI: 10.1016/0021-8693(84)90233-3
author: David Gluck
published in: Journal of Algebra
publication date: 1984-01-01T00:00:00Z
review text: A famous conjecture by R. Brauer states that the number of irreducible characters in a block of a finite group is bounded by the order of its defect group. The paper under review completes the proof of this conjecture for the following cases: (1) the group has odd order, (2) the group is solvable and the block is a 2-block. By results of Fong and Nagao the problem reduces to the situation where the group is a semidirect product GV of an odd order group G with an elementary abelian p-group V such that G acts faithfully and irreducibly on V. In this case the aim is to show that \(k(GV)\leq | V|\) where k(GV) denotes the number of conjugacy classes of GV. This is what the author calls the ''k(GV) problem''. Let W be a primitive module for a subgroup H of G such that the G-module V is isomorphic to the induced module \(W^ G\). Earlier results by \textit{R. Knörr} [Ill. J. Math. 28, 181-210 (1984; reviewed above)] and the author [Can. J. Math. 35, 59-67 (1983; Zbl 0509.20002)] settle the cases where W has order p or the Fitting subgroup of \(H/C_ H(W)\) is abelian. In this paper the proof is obtained for the remaining case. The main tools are methods coming from Knörr's Habilitationsschrift and a result of \textit{T. Wolf} [Can. J. Math. 34, 1097-1111 (1982; Zbl 0476.20028)] on solvable completely reducible linear groups. The main theorem of this paper has also been obtained by R. Knörr, but the problem for arbitrary solvable groups seems to be open.
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 3851354
zbMATH Keywords: number of irreducible characters
zbMATH Keywords: block
zbMATH Keywords: defect group
zbMATH Keywords: primitive module
zbMATH Keywords: solvable groups
reviewed by: Burkhard Külshammer
MaRDI profile type: Publication
full work available at URL: 0021-8693(84)90233-3
OpenAlex ID: W2067937158
cites work: Q5340151
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description: scientific article; zbMATH DE number 3851354