1 Preliminaries

1.1 Integer Factorization


1.2 Functions


1.3 Binary Operators


1.4 Modular Arithmetic


1.5 Rational and Real Numbers

2 Understanding the Group Concept


2.1 Introduction to Groups


2.2 Modular Congruence


2.3 The Definition of a Group

3 The Structure within a Group


3.1 Generators of Groups


3.2 Defining Finite Groups in SageMath


3.3 Subgroups

4 Patterns within the Cosets of Groups


4.1 Left and Right Cosets


4.2 Writing Secret Messages


4.3 Normal Subgroups


4.4 Quotient Groups

5 Mappings between Groups


5.1 Isomorphisms


5.2 Homomorphisms


5.3 The Three Isomorphism Theorems

6 Permutation Groups


6.1 Symmetric Groups


6.2 Cycles


6.3 Cayley’s Theorem


6.4 Numbering the Permutations

7 Building Larger Groups from Smaller Groups


7.1 The Direct Product


7.2 The Fundamental Theorem of Finite Abelian Groups


7.3 Automorphisms


7.4 Semi-Direct Products

8 The Search for Normal Subgroups


8.1 The Center of a Group


8.2 The Normalizer and Normal Closure Subgroups


8.3 Conjugacy Classes and Simple Groups


8.4 Subnormal Series and the Jordan-Holder Theorem


8.5 Solving the Pyraminx™

9 Introduction to Rings


9.1 The Definition of a Ring


9.2 Entering Finite Rings into SageMath


9.3 Some Properties of Rings

10 The Structure within Rings


10.1 Subrings


10.2 Quotient Rings and Ideals


10.3 Ring Isomorphisms


10.4 Homomorphisms and Kernels

11 Integral Domains and Fields


11.1 Polynomial Rings


11.2 The Field of Quotients


11.3 Complex Numbers