Ebook: The Theory of Classes of Groups

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.

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This book gives a systematic introduction to the theory of classes of groups, including research subjects, major (recent) research achievements, and directions for future research. It clearly and concisely treats a wealth of topics, such as a brief introduction to the fundamental knowledge of group theory; the classical part of the theory of classes of groups covering mainly F-covering subgroups, F-projectors, F-injectors and F-normalisers; local formations; Schunck classes; Fitting classes; properties of local formations; formation constructions of finite groups and related applications; and the algebra of formations.
Audience: This volume will be of interest to mathematicians involved in group theory and generalisations, algebras, order, lattices, ordered algebraic structures, general mathematical systems and the mathematics of physics and chemistry.

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This book gives a systematic introduction to the theory of classes of groups, including research subjects, major (recent) research achievements, and directions for future research. It clearly and concisely treats a wealth of topics, such as a brief introduction to the fundamental knowledge of group theory; the classical part of the theory of classes of groups covering mainly F-covering subgroups, F-projectors, F-injectors and F-normalisers; local formations; Schunck classes; Fitting classes; properties of local formations; formation constructions of finite groups and related applications; and the algebra of formations.
Audience: This volume will be of interest to mathematicians involved in group theory and generalisations, algebras, order, lattices, ordered algebraic structures, general mathematical systems and the mathematics of physics and chemistry.
Content:
Front Matter....Pages i-xi
Fundamentals of the Theory of Finite Groups....Pages 1-57
Classical F-Subgroups....Pages 58-95
Formation Structures of Finite Groups....Pages 96-166
Algebra of Formations....Pages 167-224
Supplementary Information on Algebra and Theory of Sets....Pages 225-236
Back Matter....Pages 237-258

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This book gives a systematic introduction to the theory of classes of groups, including research subjects, major (recent) research achievements, and directions for future research. It clearly and concisely treats a wealth of topics, such as a brief introduction to the fundamental knowledge of group theory; the classical part of the theory of classes of groups covering mainly F-covering subgroups, F-projectors, F-injectors and F-normalisers; local formations; Schunck classes; Fitting classes; properties of local formations; formation constructions of finite groups and related applications; and the algebra of formations.
Audience: This volume will be of interest to mathematicians involved in group theory and generalisations, algebras, order, lattices, ordered algebraic structures, general mathematical systems and the mathematics of physics and chemistry.
Content:
Front Matter....Pages i-xi
Fundamentals of the Theory of Finite Groups....Pages 1-57
Classical F-Subgroups....Pages 58-95
Formation Structures of Finite Groups....Pages 96-166
Algebra of Formations....Pages 167-224
Supplementary Information on Algebra and Theory of Sets....Pages 225-236
Back Matter....Pages 237-258
....