The Theory of Classes of Groups

by Guo Wenbin

Overview

One of the characteristics of modern algebra is the development of new tools and concepts for exploring classes of algebraic systems, whereas the research on individual algebraic systems (e. g. , groups, rings, Lie algebras, etc. ) continues along traditional lines. The early work on classes of alge­ bras was concerned with showing that one class X of algebraic systems is actually contained in another class F. Modern research into the theory of classes was initiated in the 1930's by Birkhoff's work [1] on general varieties of algebras, and Neumann's work [1] on varieties of groups. A. I. Mal'cev made fundamental contributions to this modern development. ln his re­ ports [1, 3] of 1963 and 1966 to The Fourth All-Union Mathematics Con­ ference and to another international mathematics congress, striking the­ ories of classes of algebraic systems were presented. These were later included in his book [5]. International interest in the theory of formations of finite groups was aroused, and rapidly heated up, during this time, thanks to the work of Gaschiitz [8] in 1963, and the work of Carter and Hawkes [1] in 1967. The major topics considered were saturated formations, Fitting classes, and Schunck classes. A class of groups is called a formation if it is closed with respect to homomorphic images and subdirect products. A formation is called saturated provided that G E F whenever Gjip(G) E F.

Product Details

ISBN-13:	9789401057851
Publisher:	Springer Netherlands
Publication date:	10/23/2012
Series:	Mathematics and Its Applications , #505
Edition description:	Softcover reprint of the original 1st ed. 2000
Pages:	258
Product dimensions:	6.10(w) x 9.25(h) x 0.02(d)

Table of Contents

1 Fundamentals of the Theory of Finite Groups.- §1.1 Basic Concepts.- §1.2 Homomorphism Theorems.- §1.3 Primary Groups.- §1.4 Sylow’s Theorems.- §1.5 Automorphism Groups and Semidirect Products.- §1.6 The Jordan-Hölder Theorem.- §1.7 Soluble Groups and—-soluble Groups.- §1.8 Nilpotent Groups and—-nilpotent Groups.- §1.9 Supersoluble Groups and—-supersoluble Groups.- §1.10 Some Additional Information.- 2 Classical F-Subgroups.- §2.1 Operations on Classes of Finite Groups.- §2.2 X-covering Subgroups, X-projectors, X-injectors.- §2.3 Theorems about Existence of F-covering Subgroups and F-projectors.- §2.4 The conjugacy of F-covering Subgroups.- §2.5 The Existence amd Conjugacy of F-injectors.- §2.6 .F-normalizers.- §2.7 Some Additional Information.- 3 Formation Structures of Finite Groups.- §3.1 Methods of Constructing Local Formations.- §3.2 The Stability of Formations.- §3.3 On Complements of F-coradicals.- §3.4 Minimal Non-F-groups.- §3.5 Š-formations.- §3.6 Groups with Normalizers of Sylow Subgroups Belonging to a Given Formation.- §3.7 Groups with Normalizers of Sylow Subgroups Complemented.- §3.8 Groups with Normalizers of Sylow Subgroups Having Given Indices.- §3.9 Groups with Given Local Subgroups.- §3.10 F-subnormal Subgroups.- §3.11 Some Additional Information.- 4 Algebra of Formations.- §4.1 Free Groups and Varieties of Groups.- §4.2 Generated Formations.- §4.3 Critical Formations.- §4.4 Local Formations with Complemented Subformations.- §4.5 Some Additional Information.- 5 Supplementary Information on Algebra and Theory of Sets.- §5.1 Partially Ordered Sets and Lattices.- §5.2 Classical Algebras.- §5.3 Modules over Algebras.- Index of Subjects.- List of Symbols.