A remark on Brauer's k(B)-conjecture
type: Item
instance of: scholarly article
title: A remark on Brauer's k(B)-conjecture
zbMATH Open document ID: 0709.20010
DOI: 10.1016/0021-8693(90)90185-Q
author: Reinhard Knörr
published in: Journal of Algebra
publication date: 1990-01-01T00:00:00Z
review text: ``Forgetting about context, one could say that this paper does just one thing, namely, it constructs the square root of a generalized character.'' After this unusual opening remark the author describes the generalized character \(\delta\) of a finite group G having a ``square root'': Let G act on a finite set \(\Omega\) and let F be a finite field of characteristic prime to \(| \Omega | =n\). If \(g\in G\) has m orbits on \(\Omega\), then \(\delta (g)=| F|^{n-m}\). The motivation for the result lies in the author's significant contributions to R. Brauer's k(B)-conjecture. Its corollary is: Let the p\({}'\)-group G act faithfully on the elementary abelian p-group V. Suppose that there is a \(v\in V\) so that the restriction of V to \(C_ G(V)\) is a permutation module. Then the semidirect product GV has at most \(| V|\) conjugacy classes.
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20D60
zbMATH DE Number: 4165157
zbMATH Keywords: generalized character
zbMATH Keywords: finite group
zbMATH Keywords: orbits
zbMATH Keywords: R. Brauer's k(B)-conjecture
zbMATH Keywords: \(p^{\prime }\)-group
zbMATH Keywords: permutation module
zbMATH Keywords: semidirect product
zbMATH Keywords: conjugacy classes
Wikidata QID: Q123240217
MaRDI profile type: Publication
cites work: Q3236698
cites work: ON THE NUMBER OF IRREDUCIBLE CHARACTERS OF FINITE GROUPS IN A GIVEN BLOCK
cites work: Q3922829
cites work: On the k(GV) problem
cites work: Q3952291
cites work: On the number of characters in a \(p\)-block of a \(p\)-solvable group
cites work: Q5723973
description: scientific article; zbMATH DE number 4165157