Groups that are the product of two subgroups are of particular interest to group theorists. In what way is the structure of the product related to that of its subgroups? This monograph gives the first detailed account of the most important results that have been found about groups of this form over the past 35 years. Although the emphasis is on infinite groups, some relevant theorems about finite products of groups are also proved. The material presented will be of interest for research students and specialists in group theory. In particular, it can be used in seminars or to supplement a general group theory course. A special chapter on conjugacy and splitting theorems obtained by means of the cohomology of groups has never appeared in book form and should be of independent interest.
Readership: Postgraduate mathematicians.
Author(s): Bernhard Amberg, Silvana Franciosi, Francesco de Giovanni
Series: Oxford Mathematical Monographs
Publisher: Oxford University Press
Year: 1993
Language: English
Pages: C, xii+220, B
Cover
OXFORD MATHEMATICAL MONOGRAPHS
List of Published of this Series
Products of Groups
© B. Amberg, S. Franciosi, and F. de Giovanni, 1992
ISBN 0-19-853575-9
QA 177.A53 1992 512'.2-dc2O
LCCN 92019244
Preface
Contents
Notation
1 Elementary properties of factorized groups
1.1 The factorizer
1.2 Normalizers, indices, and chain conditions
1.3 Sylow subgroups
1.4 Existence of factorizations
2 Products of nilpotent groups
2.1 Products of abelian groups
2.2 Products of central-by-finite groups
2.3 Residually finite products of abelian-by-finite groups
2.4 The theorem of Kegel and Wielandt
2.5 The structure of a finite product of nilpotent groups
3 Products of periodic groups
3.1 An example of a non-periodic product of two periodic groups
3.2 Soluble products of periodic groups
3.3 Soluble products of groups of finite exponent
4 Products of groups of finite rank
4.1 Rank formulae
4.2 The number of generators of a finite soluble group
4.3 Factorized groups with finite Prufer rank
4.4 Soluble products of polycyclic groups
4.5 Products of a nilpotent and a polycyclic group
4.6 Soluble products of groups of finite rank
5 Splitting and conjugacy theorems
5.1 Cohomology of groups
5.2 Cohomological machinery
5.3 Splitting and conjugacy
5.4 Near splitting and near conjugacy
6 Triply factorized groups
6.1 Examples of groups with an abelian triple factorization
6.2 Lower central factors and tensor products
6.3 Groups with a nilpotent triple factorization
6.4 FC-nilpotent and FC-hypercentral groups
6.5 Groups with a supersoluble triple factorization
6.6 Trifactorized groups
7 Some further topics
7.1 The 'inside-outside' problem
7.2 The Fitting length of a soluble product of nilpotent groups
7.3 Products of an abelian and an FC-group
7.4 Products of locally cyclic groups
7.5 Subnormal subgroups of factorized groups
7.6 Groups factorized by finitely many subgroups
Bibliography
Index
Back Cover
