Nilpotent metacyclic irreducible linear groups of odd order
type: Item
instance of: scholarly article
title: Nilpotent metacyclic irreducible linear groups of odd order
zbMATH Open document ID: 0832.20007
DOI: 10.1007/BF01195538
author: L. G. Kovács
author: Hyo-Seob Sim
published in: Archiv der Mathematik
publication date: 1996-02-22T00:00:00Z
review text: Let \(\mathbb{F}\) be an arbitrary field, \(G\) a nilpotent metacyclic group of odd order, and \(G^{\text{Aut }G}\) the subgroup consisting of those elements of \(G\) which are fixed by every automorphism of \(G\). We show that the images of two faithful irreducible \(\mathbb{F}\)-representations of \(G\) are linearly isomorphic if and only if the restrictions of the representations to \(G^{\text{Aut }G}\) are equivalent. If the centre of \(G\) is cyclic and the characteristic of \(\mathbb{F}\) does not divide the order of \(G\), then the number of linear isomorphism types of the irreducible \(\mathbb{F}\)-linear groups that are abstractly isomorphic to \(G\) is precisely the number of equivalence types of faithful irreducible \(\mathbb{F}\)- representations of \(G^{\text{Aut }G}\). For groups of this kind, we also show how to calculate the order of \(G^{\text{Aut }G}\).
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 20D60
Mathematics Subject Classification ID: 20D10
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20D45
Mathematics Subject Classification ID: 20E07
zbMATH DE Number: 815691
zbMATH Keywords: nilpotent metacyclic group of odd order
zbMATH Keywords: automorphism
zbMATH Keywords: faithful irreducible \(\mathbb{F}\)-representations
zbMATH Keywords: irreducible \(\mathbb{F}\)-linear groups
zbMATH Keywords: order
MaRDI profile type: Publication
cites work: Finite soluble groups
cites work: Endliche Gruppen I
cites work: Q5622421
cites work: Eine Klasse von p-Gruppen, deren Automorphismengruppen p-Gruppen sind
cites work: Realisierung von Darstellungen endlicher nilpotenter Gruppen. (Realization of representations of finite-dimensional nilpotent groups)
cites work: Degrees of irreducible representations of metacyclic groups
cites work: Faithful irreducible representations of metacyclic groups
description: scientific article; zbMATH DE number 815691