LECTURE NOTES OF WILLIAM CHEN
# INTRODUCTION TO ABSTRACT ALGEBRAIC STRUCTURES

### Chapter 1 : GROUPS >>

### Chapter 2 : FURTHER PROPERTIES OF GROUPS >>

### Chapter 3 : FURTHER EXAMPLES OF GROUPS >>

### Chapter 4 : GROUP HOMOMORPHISMS AND ISOMORPHISMS >>

### Chapter 5 : FURTHER TOPICS ON GROUPS >>

### Chapter 6 : RINGS >>

### Chapter 7 : RING HOMOMORPHISMS AND IDEALS >>

### Chapter 8 : POLYNOMIAL RINGS >>

### Chapter 9 : FIELD EXTENSIONS >>

### Chapter 10 : UNIQUE FACTORIZATION >>

### Chapter 11 : APPLICATION TO COUNTING >>

### Chapter 12 : APPLICATION TO CODING >>

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.

To read the notes, click the links below for connection to the appropriate PDF files.

The material is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, the documents may not be kept on any information storage and retrieval system without permission from the author, unless such system is not accessible to any individuals other than its owners.

- Formal Definition
- Elementary Properties
- Subgroups
- Special Subgroups

- Order
- Lagrange's Theorem
- Cyclic Groups

- The Groups Zn*
- Permutation Groups
- Dihedral Groups

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

- Direct Product of Groups
- Geometric Interpretation of Matrix Groups

- Formal Definition
- Elementary Properties
- Subrings
- Further Properties

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

- Introduction and Elementary Properties
- Factorization Properties

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

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

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

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