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-Hölder 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
Answers to Odd-Numbered Problems
Bibliography
Index
Biography
William Paulsen is a professor of mathematics at Arkansas State University. He is the author of Abstract Algebra: An Interactive Approach (CRC Press, 2009) and has published over 15 papers in applied mathematics, one of which proves that Penrose tiles can be three-colored, thus resolving a 30-year-old open problem posed by John H. Conway. Dr. Paulsen has also programmed several new games and puzzles in Javascript and C++, including Duelling Dimensions, which was syndicated through Knight Features. He earned a PhD in mathematics from Washington University in St. Louis.
