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97 pages

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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

- Sets and Maps
- Sets
- Maps
- Equivalence Relations and Partitions
- Modular Arithmetic

- Groups
- Binary Operations
- Groups: Basic Definitions
- Examples of Groups

- Subgroups
- Definition of a Subgroup
- Cosets
- Lagrange’s Theorem

- Generators and Cyclic Groups
- Orders of Group Elements
- Generating Sets
- Cyclic Groups
- Fermat’s Little Theorem

- Mappings of Groups
- Homomorphisms
- Isomorphisms

- Normal Subgroups
- Conjugates and Normal Subgroups
- Cosets of Normal Subgroups
- Kernels of Homomorphisms

- Quotient Groups
- Products of Cosets
- Quotient Groups

- The First Isomorphism Theorem
- The First Isomorphism Theorem
- Centres and Inner Automorphisms

- Group Actions
- Actions of Groups
- The Orbit-Stabilizer Theorem

- Direct Products
- Direct Products
- Direct Products of Finite Cyclic Groups
- Properties of Direct Products

- Sylow Theory
- Primes and p-Groups
- Sylow’s Theorem

- Presentations of Groups
- Introduction to Presentations
- Alphabets and Words
- Von Dyck’s Theorem
- Finitely Generated and Finitely Presented Groups
- Dehn’s Fundamental Algorithmic Problems

- Free Groups
- Reduced Words and Free Groups
- Normal Closure
- Torsion Free Groups

- Abelian Groups
- Commutator Subgroups and Abelianisations
- Free Abelian Groups
- Finitely Generated Abelian Groups
- Generalisations of Abelian Groups

- Transforming Presentations
- Tietze Transformations
- Properties of Tietze Transformations

- Free Products
- Free Products
- A Normal Form for Free Products
- The Universal Property of Free Products
- Independence of Presentation
- Decomposability

- Free Products With Amalgamation
- Free Products with Amalgamation
- Pushouts
- Independence of Presentation

- HNN Extensions
- HNN Extensions
- Relation to Free Products with Amalgamation
- The Higman-Neumann-Neumann Embedding Theorem

- Further Reading
- Bibliography
- Index

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March 23, 2018