On characters and fixed points of coprime operator groups
type: Item
instance of: scholarly article
title: On characters and fixed points of coprime operator groups
zbMATH Open document ID: 0703.20023
DOI: 10.1016/0021-8693(90)90179-R
author: I. M. Isaacs
author: Brian Hartley
published in: Journal of Algebra
publication date: 1990-01-01T00:00:00Z
review text: Suppose that \(A\) is a finite group of operators on a finite soluble group \(G\) and that \((|G|,|A|)=1\). The main theorem (Theorem A) states that if \(|C_G(A)|\leq n\), then \(|G:F_{2\ell+1}(G)|\) is bounded by a function of \(n\) and \(|A|\), where \(\ell\) is the length of the longest chain of subgroups of \(A\). Two corollaries for locally finite groups are given. The proof of this uses a theorem of Turull and the following result on representations (Theorem B). For any finite group \(A\), there exists \(\varepsilon>0\) such that the multiplicity in \(V_A\) of any of its irreducible components is at least \(\varepsilon(\dim V)\), where \(V\) is an irreducible \(kAG\)-module, \(k\) is a field the characteristic of which does not divide \(|A|\) and \(G\) is a finite, soluble group on which \(A\) acts and \((|G|,|A|)=1\). Now suppose that \(A\) is any finite group and that the characteristic of \(k\) does not divide \(|A|\). Given \(\varepsilon>0\), let \(\Delta(\varepsilon)\) denote the class of \(kA\)-modules \(W\) having the property that the multiplicity in \(W\) of any of its irreducible components is at least \(\varepsilon(\dim W)\). To prove Theorem B, a number of results about \(\Delta(\varepsilon)\) are obtained, among them Theorem C: Given \(\varepsilon>0\), there exists \(\delta>0\) such that any tensor product of modules in \(\Delta(\varepsilon)\) lies in \(\Delta(\delta)\). In this notation, Theorem B states that \(V_A\in\Delta(\varepsilon)\); this is proved by reducing it to the case when \(k=\mathbb{C}\) and then working with characters. The crucial step is to factorize the character of \(V_A\) into a number of characters, each of modules in \(\Delta(\varepsilon_0)\), where \(\varepsilon_0=\varepsilon_0(A)\), and then to use Theorem C. The factorization theorem has the corollary that if \(\chi\) is a character of the semidirect product \(AG\) and \(\chi_A\) is irreducible, then there exists an \(A\)-group \(E\) and an irreducible character \(\psi\)…
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20F50
zbMATH DE Number: 4152686
zbMATH Keywords: fixed points of automorphisms
zbMATH Keywords: Fitting heights
zbMATH Keywords: finite groups of operators
zbMATH Keywords: finite soluble groups
zbMATH Keywords: lengths
zbMATH Keywords: chains of subgroups
zbMATH Keywords: locally finite groups
zbMATH Keywords: representations
zbMATH Keywords: irreducible components
zbMATH Keywords: multiplicities
zbMATH Keywords: tensor products of modules
zbMATH Keywords: characters
zbMATH Keywords: irreducible characters
zbMATH Keywords: direct products of extraspecial groups
zbMATH Keywords: factorization theorems
reviewed by: Norman Blackburn
MaRDI profile type: Publication
cites work: A NOTE ON FIXED-POINT-FREE ACTIONS OF FINITE GROUPS
cites work: Tensor Products of Representations of Finite Groups
cites work: Carter subgroups and Fitting heights of finite solvable groups
cites work: Q5645586
cites work: Finite soluble groups admitting an automorphism of prime power order with few fixed points
cites work: Endliche Gruppen I
cites work: Characters of Solvable and Symplectic Groups
cites work: Q4103728
cites work: Generic fixed point free action of arbitrary finite groups
cites work: Fitting height of groups and of fixed points
cites work: Fixed point free action with regular orbits.
description: scientific article; zbMATH DE number 4152686