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Essential Group Theory

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Lingua:  English
Essential Group Theory is an undergraduate mathematics text book introducing the theory of groups.
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Essential Group Theory is an undergraduate mathematics text book introducing the theory of groups. It has been aimed primarily at mathematics students but those studying related disciplines such as computer science or physics should also find it useful. The first part summarizes the important points which will be found in most first undergraduate courses in group theory in brief concise chapters.The second part of the book forms an introduction to presentations of groups.

  • Introduction
  1. Sets and Maps
    1. Sets
    2. Maps
    3. Equivalence Relations and Partitions
    4. Modular Arithmetic
  2. Groups
    1. Binary Operations
    2. Groups: Basic Definitions
    3. Examples of Groups
  3. Subgroups
    1. Definition of a Subgroup
    2. Cosets
    3. Lagrange’s Theorem
  4. Generators and Cyclic Groups
    1. Orders of Group Elements
    2. Generating Sets
    3. Cyclic Groups
    4. Fermat’s Little Theorem
  5. Mappings of Groups
    1. Homomorphisms
    2. Isomorphisms
  6. Normal Subgroups
    1. Conjugates and Normal Subgroups
    2. Cosets of Normal Subgroups
    3. Kernels of Homomorphisms
  7. Quotient Groups
    1. Products of Cosets
    2. Quotient Groups
  8. The First Isomorphism Theorem
    1. The First Isomorphism Theorem
    2. Centres and Inner Automorphisms
  9. Group Actions
    1. Actions of Groups
    2. The Orbit-Stabilizer Theorem
  10. Direct Products
    1. Direct Products
    2. Direct Products of Finite Cyclic Groups
    3. Properties of Direct Products
  11. Sylow Theory
    1. Primes and p-Groups
    2. Sylow’s Theorem
  12. Presentations of Groups
    1. Introduction to Presentations
    2. Alphabets and Words
    3. Von Dyck’s Theorem
    4. Finitely Generated and Finitely Presented Groups
    5. Dehn’s Fundamental Algorithmic Problems
  13. Free Groups
    1. Reduced Words and Free Groups
    2. Normal Closure
    3. Torsion Free Groups
  14. Abelian Groups
    1. Commutator Subgroups and Abelianisations
    2. Free Abelian Groups
    3. Finitely Generated Abelian Groups
    4. Generalisations of Abelian Groups
  15. Transforming Presentations
    1. Tietze Transformations
    2. Properties of Tietze Transformations
  16. Free Products
    1. Free Products
    2. A Normal Form for Free Products
    3. The Universal Property of Free Products
    4. Independence of Presentation
    5. Decomposability
  17. Free Products With Amalgamation
    1. Free Products with Amalgamation
    2. Pushouts
    3. Independence of Presentation
  18. HNN Extensions
    1. HNN Extensions
    2. Relation to Free Products with Amalgamation
    3. The Higman-Neumann-Neumann Embedding Theorem
  19. Further Reading
  20. Bibliography
  21. Index
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Michael Batty

Michael Batty was born in 1970 in Durham, England. He gained his Ph.D. in mathematics (geometric group theory) from Warwick University in 1998. Having spent many years in academia researching and teaching mathematics and computer science, he now works as a server-side web developer. His research interests include group theory and algorithms, and he is also a keen artist in his spare time.