Description
This textbook is an introduction to group theory for upper-level undergraduates with prior experience reading and writing mathematical proofs. It covers all of the standard topics expected of an introductory text, including cyclic groups, quotient groups, the isomorphism theorems, the structure theorem for finitely generated abelian groups, group actions, and the Sylow Theorems. Throughout the book, abstract ideas are motivated by concrete examples and applications drawn from geometry, number theory, art, puzzles, and coding theory. In addition to the standard curriculum, topics include the classification of isometries of the Euclidean plane, frieze and wallpaper groups, Burnside’s Counting Theorem and its application to the art of Sol LeWitt, the mathematics of the Rubik’s Cube, finite fields and coding theory, and Dickson’s classification of the natural numbers for which every group of that order is abelian or cyclic.
Drawing on more than a decade of classroom experience, the text places examples at the center of learning, using them to build clarity, motivation, and historical perspective. Computation, proof, and conceptual understanding are integrated throughout. Each chapter concludes with suggestions for further reading and a short biography of an influential mathematician, situating the development of the field within its human and historical context.
Key Features
Flexible Course Design. A focused core supplemented by optional chapters allows instructors to tailor the course’s emphasis, whether geometric, number-theoretic, structural, or aligned with student interests.
Example-Driven Approach. More than 250 carefully chosen examples guide students from concrete computations to abstract concepts, making proofs and definitions feel natural rather than imposed.
Concepts Built Through Practice. Over 500 exercises, ranging from computational warm-ups to multi-step proofs and exploratory investigations, support sustained engagement and help students develop conceptual insight through practice.
Historical Context Through Biography. Each chapter includes a biography connecting the mathematics to the people who shaped its development.
Table of Contents
Chapter 1: Motivation
Chapter 2: Number Theory
Chapter 3: What is a Group?
Chapter 4: Important Families of Groups
Chapter 5: Lagrange’s Theorem and Cauchy’s Theorem
Chapter 6: Quotient Groups
Chapter 7: The Isomorphism Theorems
Chapter 8: The Structure Theorem for Finitely Generated Abelian Groups
Chapter 9: Divisible and Torsion Groups
Chapter 10: Groups Acting on Sets
Chapter 11: Burnside’s Counting Theorem
Chapter 12: The Sylow Theorems
Chapter 13: Geometric Group Actions
Chapter 14: Frieze Groups and Wallpaper Groups
Chapter 15: Semidirect Products
Chapter 16: When is Every Group of Order n…?
Chapter 17: The Rubik’s Cube
Chapter 18: Rings, Fields, and Algebraic Coding Theory
Appendix A: Proof Methods
Appendix B: Linear Algebra Background
Appendix C: Zorn’s Lemma and its Uses
Publication details
Publisher: CRC Press
Publication date: forthcoming
ISBN: to be announced
Purchase
Buy from CRC Press
Request an examination copy
Ask your library to order a copy