The Theory of Groups

by Hans J. Zassenhaus

Overview

Group theory represents one of the most fundamental elements of mathematics. Indispensable in nearly every branch of the field, concepts from the theory of groups also have important applications beyond mathematics, in such areas as quantum mechanics and crystallography.
Hans J. Zassenhaus, a pioneer in the study of group theory, has designed this useful, well-written, graduate-level text to acquaint the reader with group-theoretic methods and to demonstrate their usefulness as tools in the solution of mathematical and physical problems. Starting with an exposition of the fundamental concepts of group theory, including an investigation of axioms, the calculus of complexes, and a theorem of Frobenius, the author moves on to a detailed investigation of the concept of homomorphic mapping, along with an examination of the structure and construction of composite groups from simple components. The elements of the theory of p-groups receive a coherent treatment, and the volume concludes with an explanation of a method by which solvable factor groups may be split off from a finite group.
Many of the proofs in the text are shorter and more transparent than the usual, older ones, and a series of helpful appendixes presents material new to this edition. This material includes an account of the connections between lattice theory and group theory, and many advanced exercises illustrating both lattice-theoretical ideas and the extension of group-theoretical concepts to multiplicative domains.

Product Details

ISBN-13:	9780486165684
Publisher:	Dover Publications
Publication date:	08/19/2013
Series:	Dover Books on Mathematics
Sold by:	Barnes & Noble
Format:	eBook
Pages:	288
File size:	18 MB
Note:	This product may take a few minutes to download.

Table of Contents

Preface to First Edition; Preface to Second Edition
I. Elements of Group Theory
1. The Axioms of Group Theory
2. Permutation Groups
3. Investigation of Axioms
4. Subgroups
5. Cyclic Groups
6. Finite Rotation Groups
7. Calculus of Complexes
8. The Concept of Normal Subgroup
9. Normalizer, Class Equation
10. A Theorem of Frobenius
II. The Concept of Homomorphy and Groups with Operators
1. Homomorphisms
2. Representation of Groups by Means of Permutations
3. Operators and Operator Homomorphies
4. On the Automorphisms of a Group
5. Normal Chains and Normal Series
6. Commutator Groups and Commutator Forms
7. On the Groups of an Algebra
III. The Structure and Construction of Composite Groups
1. Direct Products
2. Theorems on Direct Products
3. Abelian Groups
4. Basis Theorem for Abelian Groups
5. On the Order Ideal
6. Extension Theory
7. Extensions with Cyclic Factor Group
8. Extensions with Abelian Factor Group
9. Splitting Groups
IV. Sylow p-Groups and p-Groups
1. The Sylow Theorem
2. Theorems on Sylow p-Groups
3. On p-Groups
4. On the Enumeration Theorems of the Theory of p-Groups
5. On the Descending Central Series
6. Hamiltonian Groups
7. Applications of Extension Theory
V. Transfers into a Subgroup
1. Monomial Representation and Transfers into a Subgroup
2. The Theorems of Burnside and Grün
3. Groups whose Sylow Groups are All Cyclic
4. The Principal Ideal Theorem
Appendix A. Further Exercises for Chap. II
Appendix B. Structure Theory and Direct Products. A Treatment of Chap. III, §2 on the Lattice-Theoretical Level
Appendix C. Free Products and Groups Given by a Set of Generators and a System of Defining Relations
Appendix D. Further Exercises for Chap. III
Appendix E. Further Exercises for Chap. IV, §5
Appendix F. Further Exercises for Chap. IV, §1
Appendix G. A Theorem of Wielandt. An Addendum to Chap. IV
Appendix H. Further Exercises for Chap. V, §1
Frequently Used Symbols; Bibliography; Author Index; Index

Editorial Reviews

This graduate-level text introduces group-theoretic methods, and demonstrates their usefulness as tools in the solution of problems. An unabridged reprint of the second edition published in 1958 by Chelsea. Annotation c. Book News, Inc., Portland, OR (booknews.com)

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