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.
Author(s): Hans J. Zassenhaus
Series: Dover Books on Mathematics
Edition: 2
Publisher: Dover Publications
Year: 1999
Language: English
Pages: 292
City: Mineola, New York
Tags: Group Theory
CONTENTS
PREFACE TO FIRST EDITION................................. v
PREFACE TO SECOND EDITION................................ vii
I. ELEMENTS OF CROUP THEORY
§1. The Axioms of Group Theory............................ 1
§2. Permutation Groups.................................... 4
§3. Investigation of Axioms............................... 9
§4. Subgroups............................................. 10
§5. Cyclic Groups......................................... 15
§6. Finite Rotation Groups................................ 16
§7. Calculus of Complexes................................. 19
§8. The Concept of Normal Subgroup........................ 23
§9. Normalizer,Class Equation............................. 24
§10.A Theorem of Frobenius................................ 27
II. THE CONCEPT OF HOMOMORPHY AND GROUPS WITH OPERATORS
§1. Homomorphisms........................................ 35
§2. Representation of Groups by Means of Permutations..... 39
§3. Operators and Operator Homomorphies................... 44
§4. On the Automorphisms of a Group....................... 47
§5. Normal Chains and Normal Series....................... 57
§6. Commutator Groups and Commutator Forms................ 78
§7. On the Groups of an Algebra........................... 84
III. THE STRUCTURE AND CONSTRUCTION OF COMPOSITE GROUPS
§1. Direct Products...................................... 109
§2. Theorems on Direct Products.......................... 112
§3. Abelian Groups....................................... 117
§4. Basis Theorem for Abelian Groups..................... 121
§5. On the Order Ideal................................... 123
§6. Extension Theory..................................... 124
§7. Extensions with Cyclic Factor Group.................. 128
§8. Extensions with Abelian Factor Group................. 130
§9. Splitting-Groups..................................... 133
IV. SYLOW p-GROUPS AND p-GROUPS
§1. The Sylow Theorem.................................... 135
§2. Theorems on Sylow p-Groups........................... 138
§3. On p-Groups 139
§4. 0n the Enumeration Theorems of the Theory of p-Groups 152
§5. On the Descending Central Series..................... 155
§6. Hamiltonian Groups................................... 159
§7. Applications of Extension Theory..................... 161
V. TRANSFERS into A SUBGROUP
§1. Monomial Representation and Transfers into a Subgroup 164
§2. The Theorems of Burnside and Grun.................... 169
§3. Groups whose Sylow Groups are All Cyclic............. 174
§4. The Principal Ideal Theorem.......................... 176
APPENDIXES
A. Further Exercises for Chap.II......................... 181
B. Structure Theory and Direct Products. A Treatment of
Chap. III, § 2 on the Lattice-Theoretical Level ......... 188
C. Free Products and Groups Given by a Set of Generators and
a System of Defining Relations............................ 217
D. Further Exercises for Chap. III....................... 230
E. Further Exercises for Chap. IV, §5.................... 238
F. Further Exercises for Chap. IV, §1.................... 243
G. A Theorem of Wielandt. An Addendum to Chap. IV........ 245
H. Further Exercises for Chap. V, §1..................... 252
FREQUENTLY USED SYMBOLS.................................. 253
BIBLIOGRAPHY............................................. 255
AUTHORINDEX.............................................. 259
INDEX.................................................... 259
