Introduction to Abstract Algebra

Textbook: Introduction to Higher Mathematics by Patrick Keef and David Guichard

https://www.whitman.edu/mathematics/higher_math_online/

1 Logic

1. Logical Operations
2. Quantifiers
3. De Morgan’s Laws
4. Mixed Quantifiers
5. Logic and Sets

2 Proofs

1. Direct Proofs
2. Divisibility
3. Existence proofs
4. Induction
5. Uniqueness Arguments

3 Number Theory

1. Congruence
2. 𝑍𝑛
3. The Euclidean Algorithm
4. 𝑈𝑛
5. The Fundamental Theorem of Arithmetic
6. The GCD and the LCM
7. The Chinese Remainder Theorem
8. The Euler Phi Function

4 Functions

1. Definition and Examples
2. Induced Set Functions
3. Injections and Surjections
4. More Properties of Injections and Surjections
5. Pseudo-Inverses
6. Bijections and Inverse Functions
7. Cardinality and Countability
8. Uncountability of the Reals
9. The Schröder-Bernstein Theorem
10. Cantor’s Theorem

5 Relations

1. Equivalence Relations
2. Factoring Functions
3. Ordered Sets
4. New Orders from Old
5. Partial Orders and Power Sets
6. Countable total orders

Textbook: Abstract Algebra Theory and Applications by Thomas W. Judson

https://www.math.colostate.edu/~pries/467/Judson12.pdf

Chapter 2 Integers

2.1 Mathematical Induction

2.2 The Division Algorithm

Chapter 3 Groups

3.1 Integer Equivalence Classes and Symmetries

3.2 Definitions and Examples

3.3 Subgroups

Chapter 9 Isomorphisms

9.1 Definition and Examples

Chapter 11 Homomorphisms

11.1 Group Homomorphisms

11.2 The Isomorphism Theorems

Chapter 16 Rings

16.1 Rings

16.2 Integral Domains and Fields

6.3 Ring Homomorphisms and Ideals