Groups with finitely many normalizers of non-Abelian subgroups.
type: Item
instance of: scholarly article
title: Groups with finitely many normalizers of non-Abelian subgroups.
zbMATH Open document ID: 1119.20042
author: Fausto De Mari
author: F. de Giovanni
published in: Ricerche di Matematica
publication date: 2007-07-19T00:00:00Z
review text: Recall that a group \(G\) is said to be `metahamiltonian' if all its non-Abelian subgroups are normal. These groups were introduced and investigated by G. M. Romalis and N. F. Sesekin in a series of articles published about forty years ago. They proved, among others results, that locally soluble metahamiltonian groups have finite commutator subgroup. In the article under review the authors consider the class of groups with finitely many normalizers of non-Abelian subgroups. Many results are known which show that the behaviour of normalizers has a strong influence on the structure of groups. For instance, a famous theorem of B. H. Neumann (1955), improved by I. I. Eremin (1959), states that a group \(G\) is central-by-finite iff the normalizer of every (Abelian) subgroup of \(G\) has finite index. More recently, Ya. D. Polovickiĭ (1980) has shown that a group with finite conjugacy classes is central-by-finite if it has finitely many normalizers of infinite Abelian subgroups. In particular, a group has finitely many normalizers of Abelian subgroups iff it is central-by-finite. The main results obtained in this article are the following (recall that a group \(G\) is said to be `locally graded' if every finitely generated non-trivial subgroup of \(G\) contains a proper subgroup of finite index). Theorem A: Let \(G\) be a locally graded group with finitely many normalizers of non-Abelian subgroups. Then the commutator subgroup of \(G\) is finite. Theorem B: Let \(G\) be a locally graded group with finitely many normalizers of infinite non-Abelian subgroups. Then either \(G\) is a Chernikov group or its commutator subgroup is finite. Finally, an example is given of a nilpotent group with finitely many normalizers of non-Abelian subgroups, which is neither a Chernikov group nor metahamiltonian.
reviewed by: Alessio Russo
Mathematics Subject Classification ID: 20F24
Mathematics Subject Classification ID: 20E25
Mathematics Subject Classification ID: 20E07
Mathematics Subject Classification ID: 20F22
zbMATH DE Number: 5173069
zbMATH Keywords: normalizers
zbMATH Keywords: metahamiltonian groups
zbMATH Keywords: normal non-Abelian subgroups
zbMATH Keywords: locally soluble groups
zbMATH Keywords: finite commutator subgroup
zbMATH Keywords: finite conjugacy classes
zbMATH Keywords: subgroups of finite index
zbMATH Keywords: locally graded groups
zbMATH Keywords: Chernikov groups
MaRDI profile type: Publication
full work available at URL: s11587-006-0018-6
OpenAlex ID: W2068930630
cites work: Q3765968
cites work: A special class of three Engel groups
cites work: Groups with finite classes of conjugate subgroups
cites work: Q5654235
cites work: Q4120740
cites work: Q4087414
cites work: Q3957128
cites work: Q4083615
DOI: 10.1007/S11587-006-0018-6
description: scientific article; zbMATH DE number 5173069