How to Think about Abstract Algebra provides an engaging and readable introduction to its subject, which encompasses group theory and ring theory. Abstract Algebra is central in most undergraduate mathematics degrees, and it captures regularities that appear across diverse mathematical
structures - many people find it beautiful for this reason. But its abstraction can make its central ideas hard to grasp, and even the best students might find that they can follow some of the reasoning without really understanding what it is all about.
This book aims to solve that problem. It is not like other Abstract Algebra texts and is not a textbook containing standard content. Rather, it is designed to be read before starting an Abstract Algebra course, or as a companion text once a course has begun. It builds up key information on five
topics: binary operations, groups, quotient groups, isomorphisms and homomorphisms, and rings. It provides numerous examples, tables and diagrams, and its explanations are informed by research in mathematics education.
The book also provides study advice focused on the skills that students need in order to learn successfully in their own Abstract Algebra courses. It explains how to interact productively with axioms, definitions, theorems and proofs, and how research in psychology should inform our beliefs about
effective learning.
Author(s): Lara Alcock
Edition: 1
Publisher: Oxford University Press
Year: 2021
Language: English
Commentary: Vector PDF
Pages: 320
City: Oxford, UK
Tags: Mathematics; Elementary; Abstract Algebra
Cover
How To Think About Abstract Algebra
Copyright
Preface
Contents
Symbols
Introduction
Part 1: Studying Abstract Algebra
Chapter 1: What is Abstract Algebra?
1.1 What is abstract about Abstract Algebra?
1.2 What is algebraic about Abstract Algebra?
1.3 Approaches to Abstract Algebra
Chapter 2: Axioms and Definitions
2.1 Mathematical axioms and definitions
2.2 Relating definitions to examples
2.3 The definition of group
2.4 Commutativity and rings
2.5 Mathematical objects and notation
Chapter 3: Theorems and Proofs
3.1 Theorems and proofs in Abstract Algebra
3.2 Logic in familiar algebra
3.3 Modular arithmetic
3.4 Equivalence classes
3.5 Logic in theorems
3.6 Self-explanation training
Self-explanation training
How to self-explain
Example self-explanations
Self-explanations compared with other comments
3.7 Writing proofs
Chapter 4: Studying Abstract Algebra
4.1 Who are you as a student?
4.2 Myths about learning
4.3 Effective learning
Part 2: Topics in Abstract Algebra
Chapter 5: Binary Operations
5.1 What is a binary operation?
5.2 Associativity and commutativity
5.3 Modular arithmetic
5.4 Binary operations on functions
5.5 Matrices and transformations
5.6 Symmetries and permutations
5.7 Binary operations as functions
Chapter 6: Groups and Subgroups
6.1 What is a group?
6.2 What is a subgroup?
6.3 Cyclic groups and subgroups
6.4 Cyclic subgroups and generators
6.5 Theorems about cyclic groups
6.6 Groups of familiar objects
6.7 The dihedral group D3
6.8 More symmetry groups
6.9 Permutation groups
6.10 Identifying and defining subgroups
6.11 Small groups
Chapter 7: Quotient Groups
7.1 What is a quotient group?
7.2 Quotient groups in cyclic groups
7.3 Element–coset commutativity
7.4 Left and right cosets
7.5 Normal subgroups: theory
7.6 Normal subgroups: examples
7.7 Lagrange’s Theorem
Chapter 8: Isomorphisms and Homomorphisms
8.1 What is an isomorphism?
8.2 Isomorphism definition
8.3 Early isomorphism theory
8.4 Example isomorphisms
8.5 Isomorphic or not?
8.6 Homomorphisms
8.7 The First Isomorphism Theorem
Chapter 9: Rings
9.1 What is a ring?
9.2 Examples of rings
9.3 Simple ring theorems
9.4 Rings, integral domains and fields
9.5 Units, zero divisors and equations
9.6 Subrings and ideals
9.7 Ideals, quotient rings and ring homomorphisms
Conclusion
Bibliography
Index
