Groups with finitely many non-normal subgroups
type: Item
instance of: scholarly article
title: Groups with finitely many non-normal subgroups
zbMATH Open document ID: 0668.20029
DOI: 10.1007/BF01188515
author: N. S. Hekster
author: H. W. jun. Lenstra
published in: Archiv der Mathematik
publication date: 1990-01-01T00:00:00Z
review text: In this paper all groups \(G\) are classified that have only finitely many non-normal subgroups. It is proved that a group \(G\) has this property if and only if it is either Abelian, or Hamiltonian, or finite, or of the form \(A\times B\), where \(A, B\) are as follows: \(B\) contains for some prime number \(p\), a central subgroup \(C\) that is isomorphic to the group of complex roots of unity of \(p\)-power order, such that \(B/C\) is a finite Abelian \(p\)-group, and \(A\) is a finite Abelian or Hamiltonian group of order not divisible by \(p\). It follows from this result that any infinite group that has non-normal subgroups at all has at least six of them, with equality occurring for precisely one group, up to isomorphism. As an auxiliary result, all infinite Abelian groups \(G\) are described that have an automorphism that fixes almost all but not all subgroups of \(G\).
Mathematics Subject Classification ID: 20E34
Mathematics Subject Classification ID: 20E36
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20F24
Mathematics Subject Classification ID: 20K30
zbMATH DE Number: 4093750
zbMATH Keywords: finitely many non-normal subgroups
zbMATH Keywords: abelian groups
zbMATH Keywords: Hamiltonian groups
zbMATH Keywords: finite groups
zbMATH Keywords: central subgroup
zbMATH Keywords: infinite abelian groups
zbMATH Keywords: automorphism
MaRDI profile type: Publication
cites work: Groups Covered By Permutable Subsets
cites work: Q5654235
description: scientific article; zbMATH DE number 4093750