LECTURE NOTES OF WILLIAM CHEN

 

INTRODUCTION TO ABSTRACT ALGEBRAIC STRUCTURES

This set of notes has been organized in such a way to create a single volume suitable for a very brief introduction to the theory groups, rings and fields, as well as their applications to counting and coding.

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Chapter 1 : GROUPS >>

• Formal Definition
• Elementary Properties
• Subgroups
• Special Subgroups

Chapter 2 : FURTHER PROPERTIES OF GROUPS >>

• Order
• Lagrange's Theorem
• Cyclic Groups

Chapter 3 : FURTHER EXAMPLES OF GROUPS >>

• The Groups Zn*
• Permutation Groups
• Dihedral Groups

Chapter 4 : GROUP HOMOMORPHISMS AND ISOMORPHISMS >>

• Formal Definition
• Some Properties of Homomorphisms
• Normal Subgroups
• Cosets and Factor Groups
• The Fundamental Theorem of Group Homomorphisms

Chapter 5 : FURTHER TOPICS ON GROUPS >>

• Direct Product of Groups
• Geometric Interpretation of Matrix Groups

Chapter 6 : RINGS >>

• Formal Definition
• Elementary Properties
• Subrings
• Further Properties

Chapter 7 : RING HOMOMORPHISMS AND IDEALS >>

• Ring Homomorphisms
• Ideals
• Quotient Rings
• Fundamental Theorem of Ring Homomorphisms
• Prime and Maximal Ideals

Chapter 8 : POLYNOMIAL RINGS >>

• Introduction and Elementary Properties
• Factorization Properties

Chapter 9 : FIELD EXTENSIONS >>

• Ideals in Polynomial Rings
• The Structure of an Extension Field
• Characteristic of a Field
• Finite Fields
• Connection with Group Theory

Chapter 10 : UNIQUE FACTORIZATION >>

• A Simple Case
• Principal Ideal Domains
• The Simple Case Again
• Euclidean Domains
• A Shortcut
• Two Remarks

Chapter 11 : APPLICATION TO COUNTING >>

• Group Actions
• Burnside's Theorem
• Conjugates
• The Class Equation

Chapter 12 : APPLICATION TO CODING >>

• Introduction
• Group Codes
• Matrix Codes
• Error Detection and Correction
• Decoding in Matrix Codes
• Coset Leaders
• Hamming Codes
• Polynomial Codes
• Connection with Field Theory