Abstract Algebra: An Interactive Approach

“Abstract Algebra: An Interactive Approach” is a new concept in learning modern algebra. Although all the expected topics are covered thoroughly and in the most popular order, the text does offer much flexibility. Perhaps more significantly, it gives professors and students the option of including technology in the course. Each chapter in the textbook has a corresponding interactive Mathematica notebook and an interactive SageMath workbook which can be used either in the classroom, or outsideof the classroom. Students will be able to visualize the important abstract concepts, such as groups and rings (by displaying multiplication tables), homomorphisms (by showing a line graph between two groups), and permutations. This in turn allows the students to learn these difficult concepts much more quickly and obtain a firmer grasp than with a traditional textbook. Thus, the colorful diagrams produced by Mathematica give added value to the students. Teachers can run the Mathematica or SageMath notebooks in the classroom to have the studentvisualizethedynamicsofgroupsandrings.Studentshavetheoptionofrunning the notebooks at home, and experiment with different groups or rings.Some of the exercises require technology, but most of the exercises are of the standard type, with various difficulty levels. The third edition is focused for use in an undergraduate single semester course and reduces the breadth of coverage, size, and cost of the previous editions. Additional changes include: § Binary operators are now in an independent section. § The extended Euclidean algorithm is included. § Many more homework problems are added to some sections § Mathematical induction is moved to section 1.2. In spite of the additional software emphasis, the text is not short on rigor.All of the classical proofs are included, although some of the harder proofs can be shortened by using technology.

Table of Contents:
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