Dual modules and group actions on extra-special groups
type: Item
instance of: scholarly article
title: Dual modules and group actions on extra-special groups
zbMATH Open document ID: 0742.20010
DOI: 10.1216/RMJ-1988-18-3-505
author: I. M. Isaacs
published in: Rocky Mountain Journal of Mathematics
publication date: 1992-06-25T00:00:00Z
review text: The author shows how to construct certain examples of groups with a normal extra-special subgroup. Such examples are useful in character, representation, and solvable group theory. Let \(G\) be a normal subgroup of a group \(H\) of index two. Suppose that \(V\) is an irreducible \(FG\)-module over a field \(F=GF(p)\) for a prime \(p\). Suppose that \(V\) is conjugate to its dual module \(V^ \wedge\) in \(H\). That is, the \(FH\)-module \(W\) induced by \(V\) contains copies of both \(V\) and \(V^ \wedge\). Then there is an extra-special group \(E\) on which \(H\) acts such that (a) \(E/Z(E)\) is isomorphic with \(W\), (b) with this isomorphism, the inverse image in \(E\) of \(V\otimes 1\) in \(E\) is abelian, and (c) \(G\) acts trivially on \(Z(E)\) and \(H\backslash G\) either inverts or is trivial on \(Z(E)\). In this last case, \(E\) may be chosen to satisfy either condition except when an absolutely irreducible constituent of \(V\) is \(H\)-conjugate to its contragredient. Viewing \(F\) as a trivial module, \(V^ \wedge\) acts upon \(V\oplus F\). The semidirect product \((V\oplus F)V^ \wedge\) admits \(G\) and is \(E\). Since \(W\) is isomorphic with \(V\oplus V^ \wedge\) as an \(FG\)-module, and since \(H\) acts on \(E/Z(E)\) as it does on \(W\), the proof consists of an examination of how action of \(H\) might lift from \(E/Z(E)\) to \(E\). Form primitive modules may be divided into three kinds [the reviewer, On the structure of a representation of a finite solvable group (1985; Zbl 0562.20002), Theorem 4.12]. This example sheds light on the second two kinds.
Mathematics Subject Classification ID: 20C05
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 3485
zbMATH Keywords: normal extra-special subgroup
zbMATH Keywords: solvable group
zbMATH Keywords: irreducible \(FG\)-module
zbMATH Keywords: dual module
MaRDI profile type: Publication
full work available at URL: rmj-1988-18-3-505
OpenAlex ID: W2139648744
description: scientific article; zbMATH DE number 3485