Zeros of Brauer characters and linear actions of finite groups.
type: Item
instance of: scholarly article
title: Zeros of Brauer characters and linear actions of finite groups.
zbMATH Open document ID: 1253.20007
author: Silvio Dolfi
author: Emanuele Pacifici
published in: Journal of Algebra
publication date: 2012-02-11T00:00:00Z
review text: Let \(G\) be a finite group, and \(p\) a prime number greater than \(3\). It is known that, if every irreducible \(p\)-Brauer character of \(G\) does not vanish on any \(p'\)-element of \(G\), then \(G\) is solvable. The primary aim of this work is to describe the structure of groups satisfying the above condition; among other more specific properties, it is shown that the \(p'\)-length of \(G\) is at most 2 (the bound being the best possible). The structural results are obtained as an application of the crucial theorem in this paper, that deals with particular linear actions of solvable groups on finite vector spaces. That theorem runs as follows: Theorem. Let \(G\) be a solvable group, \(q\) a prime number and \(p\) another prime number with \(p\geq 5\). Let \(V\) be a faithful primitive \(G\)-module with \(\#V=q^n\). Assume that \(p\) does not divide the order of \(G\) and that for each orbit for the action of \(G\) on \(V\) there exist an element in such an orbit which is centralized by some \(p'\)-element of \(G\). Then the following conclusions hold: (a) \(G\cong H\), where \(H\leq\Gamma(V)\), the semilinear group acting naturally on \(V\); (b) \(G\) is a Frobenius group with cyclic kernel of order \(p^a\) and Frobenius complement of prime order \(r\) with \(r\) dividing \(n\). Furthermore, \(n\) does not divide \(p-1\) and, for every \(v\in V\setminus\{0\}\), it holds that \(\#C_G(v)=r\). Finally, \(p^a=(q^n-1)/(q^{n/r}-1)\).
Mathematics Subject Classification ID: 20C20
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20D20
zbMATH DE Number: 6005448
zbMATH Keywords: Brauer characters
zbMATH Keywords: finite groups
zbMATH Keywords: linear actions
zbMATH Keywords: vector spaces
reviewed by: Robert W. van der Waall
MaRDI profile type: Publication
full work available at URL: j.jalgebra.2011.05.023
OpenAlex ID: W2088387339
cites work: Q3922829
cites work: On independent sets in the class graph of a finite group.
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cites work: Zeros of characters of finite groups
cites work: Q4273959
cites work: Q4387775
cites work: Regular orbits of finite primitive solvable groups
DOI: 10.1016/J.JALGEBRA.2011.05.023
description: scientific article; zbMATH DE number 6005448