The quasisimple case of the \(k(GV)\)-conjecture
type: Item
instance of: scholarly article
title: The quasisimple case of the \(k(GV)\)-conjecture
zbMATH Open document ID: 0977.20002
author: Udo Riese
published in: Journal of Algebra
publication date: 2001-10-10T00:00:00Z
review text: Let \(F\) be a finite field of characteristic \(p\) and let \(G\) be a finite group of \(p'\)-order. Let \(V\) be a finite-dimensional faithful \(FG\)-module. Let \(v\) be any vector in \(V\) and let \(C_G(v)\) denote the subgroup of \(G\) that fixes \(v\). We say that \(v\) is an \(F\)-real vector if \(V\), considered as an \(FC_G(v)\)-module, is isomorphic to its dual module. We say that \(v\) is an \(F\)-weakly real vector if \(V\), considered as an \(FC_G(v)\)-module, contains a faithful submodule that is isomorphic to its dual. Interest in the existence of such real or weakly real vectors arises from their application to the solution of a special case of Brauer's \(k(B)\)-problem, concerning the number of irreducible complex characters in a \(p\)-block \(B\) of a finite \(p\)-solvable group. The author considers the problem of showing the existence of an \(F\)-real or \(F\)-weakly real vector when \(G\) contains a normal quasisimple subgroup \(E\) that acts irreducibly on \(V\). His main theorem is that in these circumstances an \(F\)-real vector exists except in three different cases, as follows. \(\bullet\) \(E\cong\text{SL}_2(5)\), \(\dim V=2\), \(|F|=11\), \(19\), or \(31\). \(\bullet\) \(E\cong 2.A_6\), \(\dim V=4\), \(|F|=7\). \(\bullet\) \(E\cong\text{Sp}_4(3)\), \(\dim V=4\), \(|F|=7\), \(13\), \(19\) or \(31\). In the third case, when \(|F|=31\), there is a weakly real vector for \(G\), but in all the other cases there is no weakly real vector. (Note that in general, \(G\) is strictly bigger than \(E\), and \(E\) may itself have real vectors.) The importance of this theorem is that, in combination with the work of several other researchers, it shows that Brauer's \(k(B)\)-conjecture is true for all \(p\)-solvable groups unless possibly \(p\) is one of \(3\), \(5\), \(7\), \(11\), \(13\), \(19\) or \(31\). The author builds on earlier work of \textit{D. P. M. Goodwin} [J. Algebra 227, No. 2, 395-432, 433-473 (2000; Zbl 0970.20005, Zbl 0970.20006)], who …
reviewed by: Rod Gow
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20C20
zbMATH DE Number: 1563171
zbMATH Keywords: Brauer's \(k(GV)\)-conjecture
zbMATH Keywords: \(k(B)\)-problem
zbMATH Keywords: quasisimple groups
zbMATH Keywords: weakly real vectors
zbMATH Keywords: faithful modules
zbMATH Keywords: numbers of characters
zbMATH Keywords: \(p\)-solvable groups
describes a project that uses: GAP
MaRDI profile type: Publication
full work available at URL: jabr.2000.8461
OpenAlex ID: W2082164436
Wikidata QID: Q123302466
cites work: Q3684278
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: Regular orbits of linear groups with an application to the \(k(GV)\)-problem. I
cites work: Regular orbits of linear groups with an application to the \(k(GV)\)-problem. II
cites work: On the Number of Characters in a Block and the k (GV ) PROBLEM FOR SELF-DUAL V
cites work: Q4312077
cites work: On the number of characters in a \(p\)-block of a \(p\)-solvable group
cites work: Regular orbits of linear groups
cites work: Group actions on Cartesian powers with applications to represenation theory.
cites work: Self-dual modules and real vectors for solvable groups
cites work: Some remarks on the \(k(GV)\) problem
cites work: Further reductions for the \(k(GV)\)-problem
cites work: On Brauer's \(k(B)\)-problem
cites work: The complex Leech lattice and maximal subgroups of the Suzuki group
DOI: 10.1006/JABR.2000.8461
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description: scientific article; zbMATH DE number 1563171