Intro
Preface
Contents
About the Authors
Symbols and Acronyms
List of Figures
List of Tables
1 Background Results in Set Theory
1.1 Operations on Sets
1.2 Principle of Mathematical Induction
1.3 Binary Relations on Sets
1.4 Types of Binary Relations on Sets
1.5 Functions
1.6 Matrices
1.7 Geometric Transformations and Symmetries in the Plane
References
2 Algebraic Operations on Integers
2.1 Basic Algebraic Operations on Integers
2.2 Divisibility of Integers
2.3 Common Divisors of Integers
2.4 Euclidean Algorithm (Euclid's Algorithm)

2.5 Bézout's Lemma (Bézout's Identity)
2.6 Relatively Prime Integers
2.7 Common Multiples of Integers
2.8 Prime Numbers and the Fundamental Theorem of Arithmetic
2.9 Applications of the Fundamental Theorem of Arithmetic
Reference
3 The Integers Modulo n
3.1 Structure of Integers Modulo n
3.2 Functions on the Integers Modulo n
3.3 Algebraic Operations the Integers Modulo n
3.4 The Addition Modulo n and Multiplication Modulo n Tables
3.5 Use of the "mod n" Formula
3.6 Linear Equations on the Integers Modulo n
Reference
4 Semigroups and Monoids

4.1 Binary Operations on Sets
4.2 Semigroups and Monoids
4.3 Invertible Elements in Monoids
4.4 Idempotent Elements in Semigroups
5 Groups
5.1 Definition and Basic Examples
5.2 Cayley's Tables for Finite Groups
5.3 Additive and Multiplicative Groups of Integers Modulo n
5.4 Abelian Groups and the Center of a Group
5.5 The Order of an Element in a Group
5.6 Direct Product of Groups
Reference
6 The Symmetric Group "An Example of Finite Nonabelian Group"
6.1 Matrix Representation of Permutations
6.2 Cycles on { 1,2, ,n }
6.3 Orbits of a Permutation

6.4 Order of a Permutation
6.5 Odd and Even Permutations
References
7 Subgroups
7.1 Definitions and Basic Examples
7.2 Operations on Subgroups
7.3 Subgroups Generated by a Set and Finitely Generated Subgroups
7.4 Cosets of Subgroups and Lagrange's Theorem
7.5 Normal Subgroups of a Group
7.6 Internal Direct Product of Subgroups
7.7 The Quotient Groups
References
8 Group Homomorphisms and Isomorphic Groups
8.1 Group Homomorphisms, Definitions, and Basic Examples
8.2 The Kernel and Image of Homomorphism
8.3 Group Isomorphisms and Cayley's Theorem

8.4 The Fundamental Theorems of Homomorphisms
8.5 Group Actions and Group Homomorphisms
Reference
9 Classification of Finite Abelian Groups
9.1 Cyclic Groups
9.2 Primary Groups
9.3 Independent Subsets, Spanning Subsets, and Bases of a Group
9.4 The Fundamental Theorem of Finite Abelian Groups
References
10 Group Theory and Sage
10.1 What Is Sage?
10.2 Examples for Using Sage in Group Theory
10.2.1 Commands Related to Sets and Basic Operations
10.2.2 Commands Related to Integers Modulo n
10.2.3 Commands Related to Groups