Contents

Preface xi

Chapter 1. Arithmetic 1

1. Integers 1

1.1. Basic Properties 1

1.2. Induction 2

1.3. Division Algorithm 4

1.4. Divisors 5

1.5. Fundamental Theorem of Arithmetic 8

1.6. Exercises 1.1–1.28 9

2. Modular Arithmetic 13

2.1. Congruence 13

2.2. Congruence Classes 14

2.3. Arithmetic 15

2.4. Structure 16

2.5. Applications 20

2.6. Equivalence Relations 24

2.7. Exercises 1.29–1.72 26

Chapter 2. Groups 33

1. Definitions and Examples 33

1.1. Binary Operations 33

1.2. Definition of a Group 34

1.3. Examples 35

1.4. Exercises 2.1–2.29 41

2. Basic Properties and Order 46

2.1. Exercises 2.30–2.52 48

3. Subgroups and Direct Products 51

3.1. Subgroups 51

3.2. Direct Products 54

3.3. Exercises 2.53–2.91 54

4. Morphisms 59

4.1. Introduction 59

4.2. Definitions and Examples 60

4.3. Basic Properties 62

4.4. Exercises 2.92–2.118 63

vii