Essential Group Theory
Publisher: Bookboon.com
Published: 2013
ISBN-13: 9788740303018
Table of contents: title: Content; title: Introduction; title: 1. Sets and Maps; title: 1.1. Sets; title: 1.2. Maps; title: 1.3. Equivalence Relations and Partitions; title: 1.4. Modular Arithmetic; title: 2. Groups; title: 2.1. Binary Operations; title: 2.2. Groups: Basic Definitions; title: 2.3. Examples of Groups; title: 3. Subgroups; title: 3.1. Definition of a Subgroup; title: 3.2. Cosets; title: 3.3. Lagrange’s Theorem; title: 4. Generators and Cyclic Groups; title: 4.1. Orders of Group Elements; title: 4.2. Generating Sets; title: 4.3. Cyclic Groups; title: 4.4. Fermat’s Little Theorem; title: 5. Mappings of Groups; title: 5.1. Homomorphisms; title: 5.2. Isomorphisms; title: 6. Normal Subgroups; title: 6.1. Conjugates and Normal Subgroups; title: 6.2. Cosets of Normal Subgroups; title: 6.3. Kernels of Homomorphisms; title: 7. Quotient Groups; title: 7.1. Products of Cosets; title: 7.2. Quotient Groups; title: 8. The First Isomorphism Theorem; title: 8.1. The First Isomorphism Theorem; title: 8.2. Centres and Inner Automorphisms; title: 9. Group Actions; title: 9.1. Actions of Groups; title: 9.2. The Orbit-Stabilizer Theorem; title: 10. Direct Products; title: 10.1. Direct Products; title: 10.2. Direct Products of Finite Cyclic Groups; title: 10.3. Properties of Direct Products; title: 11. Sylow Theory; title: 11.1. Primes and p-Groups; title: 11.2. Sylow’s Theorem; title: 12. Presentations of Groups; title: 12.1. Introduction to Presentations; title: 12.2. Alphabets and Words; title: 12.3. Von Dyck’s Theorem; title: 12.4. Finitely Generated and Finitely Presented Groups; title: 12.5. Dehn’s Fundamental Algorithmic Problems; title: 13. Free Groups; title: 13.1. Reduced Words and Free Groups; title: 13.2. Normal Closure; title: 13.3. Torsion Free Groups; title: 14. Abelian Groups; title: 14.1. Commutator Subgroups and Abelianisations; title: 14.2. Free Abelian Groups; title: 14.3. Finitely Generated Abelian Groups; title: 14.4. Generalisations of Abelian Groups; title: 15. Transforming Presentations; title: 15.1. Tietze Transformations; title: 15.2. Properties of Tietze Transformations; title: 16. Free Products; title: 16.1. Free Products; title: 16.2. A Normal Form for Free Products; title: 16.3. The Universal Property of Free Products; title: 16.4. Independence of Presentation; title: 16.5. Decomposability; title: 17. Free Products With Amalgamation; title: 17.1. Free Products with Amalgamation; title: 17.2. Pushouts; title: 17.3. Independence of Presentation; title: 18. HNN Extensions; title: 18.1. HNN Extensions; title: 18.2. Relation to Free Products with Amalgamation; title: 18.3. The Higman-Neumann-Neumann Embedding Theorem; title: 19. Further Reading; title: 20. Bibliography; title: 21. Index