Self-dual modules and real vectors for solvable groups
type: Item
instance of: scholarly article
title: Self-dual modules and real vectors for solvable groups
zbMATH Open document ID: 0968.20003
author: Udo Riese
author: Peter Schmid
published in: Journal of Algebra
publication date: 2001-06-18T00:00:00Z
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. (This is equivalent to saying that the Brauer character of \(C_G(v)\) obtained from its action on \(V\) is real-valued.) Interest in the existence of such real vectors arises in the theory of \(p\)-blocks of \(p\)-solvable groups, as we shall now explain. Suppose that \(H\) is a finite group and \(B\) is a \(p\)-block of irreducible characters of \(H\) with defect group \(D\). Brauer's \(k(B)\)-problem is to show that the number of irreducible complex characters of \(H\) in \(B\) is at most \(|D|\). To solve Brauer's \(k(B)\)-problem when \(H\) is \(p\)-solvable, it suffices to assume that \(H=NG\), where \(N\) is a minimal normal self-centralizing elementary Abelian \(p\)-subgroup of \(H\) and \(G\) is a \(p'\)-group. We may consider \(N\) to be a faithful module for \(FG\), where \(F\) is the field of order \(p\). By a theorem of \textit{G.~R.~Robinson} [J. Algebra 172, No. 1, 159-166 (1995; Zbl 0832.20006)], there is an affirmative answer to Brauer's problem if \(N\) has an \(F\)-real vector. The authors consider the problem of showing the existence of an \(F\)-real vector when \(G\) is solvable and \(F\) is any finite field. In their Theorem A, they show that, when \(G\) is solvable, \(F\)-real vectors exist unless \(|F|=3\), \(5\), \(7\) or \(13\). Generalizing earlier work of Gluck and Knörr, they also show in their Theorem B that \(F\)-real vectors exist if \(G\) has a normal \(2\)-complement or if the Fitting subgroup of \(G\) is not a \(2\)-group when \(|F|=3\) or \(5\), and not a \(\{2,3\}\)-group in general (part of this theorem is described in a note at the end of the paper, impro…
reviewed by: Rod Gow
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20D10
zbMATH DE Number: 1448641
zbMATH Keywords: Brauer's \(k(B)\)-problem
zbMATH Keywords: self-dual modules
zbMATH Keywords: finite solvable groups
zbMATH Keywords: real vectors
zbMATH Keywords: finite-dimensional faithful modules
zbMATH Keywords: Brauer characters
zbMATH Keywords: blocks
zbMATH Keywords: irreducible characters
zbMATH Keywords: defect groups
MaRDI profile type: Publication
full work available at URL: jabr.1999.8231
OpenAlex ID: W1971409039
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DOI: 10.1006/JABR.1999.8231
description: scientific article; zbMATH DE number 1448641