Fundamentals of the Theory of Groups R. G. Burns

Fundamentals of the Theory of Groups By R. G. Burns

Summary

Our intention was to set forth just the fundamentals of group theory, avoiding excessive detail and skirting the quagmire of generalizations (however a few generalizations are nonetheless considered-see the last sections of Chapters 6 and 7).

Fundamentals of the Theory of Groups by R. G. Burns

The present edition differs from the first in several places. In particular our treatment of polycyclic and locally polycyclic groups-the most natural generalizations of the classical concept of a finite soluble group-has been expanded. We thank Ju. M. Gorcakov, V. A. Curkin and V. P. Sunkov for many useful remarks. The Authors Novosibirsk, Akademgorodok, January 14, 1976. v Preface to the First Edition This book consists of notes from lectures given by the authors at Novosi birsk University from 1968 to 1970. Our intention was to set forth just the fundamentals of group theory, avoiding excessive detail and skirting the quagmire of generalizations (however a few generalizations are nonetheless considered-see the last sections of Chapters 6 and 7). We hope that the student desiring to work in the theory of groups, having become acquainted with its fundamentals from these notes, will quickly be able to proceed to the specialist literature on his chosen topic. We have striven not to cross the boundary between abstract and scholastic group theory, elucidating difficult concepts by means of simple examples wherever possible. Four types of examples accompany the theory: numbers under addition, numbers under multiplication, permutations, and matrices.

1 Definition and Most Important Subsets of a Group.- 1. Definition of a Group.- 1.1. Axioms. Isomorphism.- 1.2. Examples.- 2. Subgroups. Normal Subgroups.- 2.1. Subgroups.- 2.2. Generating Sets.- 2.3. Cyclic Subgroups.- 2.4. Cosets.- 2.5. Classes of Conjugate Elements.- 3. The Center. The Commutator Subgroup.- 3.1. The Center.- 3.2. The Commutator Subgroup.- 2 Homorphisms.- 4. Homomorphisms and Factors.- 4.1. Definitions.- 4.2. Homomorphism Theorems.- 4.3. Subcartesian Products.- 4.4. Subnormal Series.- 5. Endomorphisms. Automorphisms.- 5.1. Definitions.- 5.2. Invariant Subgroups.- 5.3. Complete Groups.- 6. Extensions by Means of Automorphisms.- 6.1. The Holomorph.- 6.2. Wreath Products.- 3 Abelian Groups.- 7. Free Abelian Groups. Rank.- 7.1. Free Abelian Groups.- 7.2. Rank of an Abelian Group.- 8. Finitely Generated Abelian Groups.- 9. Divisible Abelian Groups.- 10. Periodic Abelian Groups.- 4 Finite Groups.- 11. Sylow p-Subgroups.- 11.1. Sylow's Theorem.- 11.2. An Application to Groups of Order pq.- 11.3. Examples of Sylow p-Subgroups.- 12. Finite Simple Groups.- 12.1. The Alternating Groups.- 12.2. The Projective Special Linear Groups.- 13. Permutation Groups.- 13.1. The Regular Representation.- 13.2. Representations by Permutations of Cosets.- 13.3. Transitivity. Primitivity.- 5 Free Groups and Varieties.- 14. Free Groups.- 14.1. Definition.- 14.2. A Matrix Representation.- 14.3. Subgroups.- 14.4. The Lower Central Series and the Derived Series.- 15. Varieties.- 15.1. Laws and Varieties.- 15.2. An Alternative Approach to Varieties.- 6 Nilpotent Groups.- 16. General Properties and Examples.- 16.1. Definition.- 16.2. General Properties.- 16.3. Nilpotent Groups of Automorphisms.- 17. The Most Important Subclasses.- 17.1. Finite Nilpotent Groups.- 17.2. Finitely Generated Nilpotent Groups.- 17.3. Torsion-Free Nilpotent Groups.- 18. Generalizations of Nilpotency.- 18.1. Local Nilpotence.- 18.2. The Normalizer Condition.- 18.3. The Engel Condition.- 7 Soluble Groups.- 19. General Properties and Examples.- 19.1. Definitions.- 19.2. Soluble Groups Satisfying the Maximal Condition.- 19.3. Soluble Groups Satisfying the Minimal Condition.- 20. Finite Soluble Groups.- 20.1. Hall and Carter Subgroups.- 20.2. On the Complete Reducibility of Representations.- 20.3. A Criterion for Supersolubility.- 21. Soluble Matrix Groups.- 21.1. Almost-Triangularizability.- 21.2. The Polycyclicity of the Soluble Subgroups of GLn(Z).- 21.3. The Embeddability in GLn(Z) of the Holomorph of a Polycyclic Group.- 22. Generalizations of Solubility.- 22.1. Kuros-?ernikov Classes.- 22.2. Examples.- 22.3. The Local Theorem.- Append.- Auxiliary Results from Algebra, Logic and Number Theory.- 23. On Nilpotent Algebras.- 23.1. Nilpotence of Associative and Lie Algebras.- 23.2. Non-Nilpotent Nilalgebras.- 24. Local Theorems of Logic.- 24.1. Algebraic Systems.- 24.2. The Language of the Predicate Calculus.- 24.3. The Local Theorems.- 25. On Algebraic Integers.- Index of Notations for Classical Objects.

Fundamentals of the Theory of Groups R. G. Burns

Fundamentals of the Theory of Groups by R. G. Burns

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Fundamentals of the Theory of Groups Summary

Fundamentals of the Theory of Groups by R. G. Burns

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