How to Think About Abstract Algebra

by Lara Alcock

Overview

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.

Product Details

ISBN-13:	9780198843382
Publisher:	Oxford University Press
Publication date:	03/29/2021
Pages:	308
Product dimensions:	7.60(w) x 4.90(h) x 0.70(d)

About the Author

Lara Alcock, Reader and Head of Department, Mathematics, Education Centre, Loughborough University

Lara Alcock is a Reader in Mathematics Education at Loughborough University. She collaborates with colleagues, PhD students and project students to conduct research on mathematical thinking and learning, specializing in reasoning among undergraduate mathematics students and professional mathematicians. She is author of two previous research-informed study guides for undergraduate students: How to Study for a Mathematics Degree and How to Think about Analysis. She also authored the popular mathematics book Mathematics Rebooted: A Fresh Approach to Understanding.

Table of Contents

Symbols xiii

Introduction xv

Part 1 Studying Abstract Algebra

1 What is Abstract Algebra? 3

1.1 What is abstract about Abstract Algebra? 3

1.2 What is algebraic about Abstract Algebra? 4

1.3 Approaches to Abstract Algebra 7

2 Axioms and Definitions 13

2.1 Mathematical axioms and definitions 13

2.2 Relating definitions to examples 17

2.3 The definition of group 21

2.4 Commutativity and rings 25

2.5 Mathematical objects and notation 28

3 Theorems and Proofs 31

3.1 Theorems and proofs in Abstract Algebra 31

3.2 Logic in familiar algebra 35

3.3 Modular arithmetic 39

3.4 Equivalence classes 44

3.5 Logic in theorems 51

3.6 Self-explanation training 56

3.7 Writing proofs 60

4 Studying Abstract Algebra 67

4.1 Who are you as a student? 67

4.2 Myths about learning 68

4.3 Effective learning 74

Part 2 Topics in Abstract Algebra

5 Binary Operations 79

5.1 What is a binary operation? 79

5.2 Associativity and commutativity 81

5.3 Modular arithmetic 84

5.4 Binary operations on functions 91

5.5 Matrices and transformations 93

5.6 Symmetries and permutations 96

5.7 Binary operations as functions 101

6 Groups and Subgroups 105

6.1 What is a group? 105

6.2 What is a subgroup? 108

6.3 Cyclic groups and subgroups 112

6.4 Cyclic subgroups and generators 116

6.5 Theorems about cyclic groups 119

6.6 Groups of familiar objects 123

6.7 The dihedral group D₃ 127

6.8 More symmetry groups 132

6.9 Permutation groups 136

6.10 Identifying and defining subgroups 141

6.11 Small groups 145

7 Quotient Groups 151

7.1 What is a quotient group? 151

7.2 Quotient groups in cyclic groups 154

7.3 Element-coset commutativity 158

7.4 Left and right cosets 163

7.5 Normal subgroups: theory 167

7.6 Normal subgroups: examples 171

7.7 Lagrange's Theorem 176

8 Isomorphisms and Homomorphisms 183

8.1 What is an isomorphism? 183

8.2 Isomorphism definition 186

8.3 Early isomorphism theory 192

8.4 Example isomorphisms 197

8.5 Isomorphic or not? 202

8.6 Homomorphisms 205

8.7 The First Isomorphism Theorem 212

9 Rings 219

9.1 What is a ring? 219

9.2 Examples of rings 222

9.3 Simple ring theorems 228

9.4 Rings, integral domains and fields 231

9.5 Units, zero divisors and equations 235

9.6 Subrings and ideals 242

9.7 Ideals, quotient rings and ring homomorphisms 248

Conclusion 257

Bibliography 263

Index 277

Editorial Reviews

"I'd very strongly recommend it to undergraduates studying maths, Sixth formers about to study maths, and anyone who did a maths degree a while ago and wants to revisit groups, rings and fields. I also recommend that any first year pure maths lecturers reading this should add this book to their course's reading list." — Chalkdust

From the Publisher