Description
This book offers a focused, modern introduction to undergraduate group theory for students who have completed linear algebra and an introductory proof-writing course. Rather than presenting the subject as a sequence of abstract definitions, it develops group theory as an interconnected story emerging from geometry, number theory, and the mathematics of symmetry. Standard topics are complemented by chapters on Euclidean isometries, divisible groups, and a generalization of Wilson’s Theorem, giving students a sense of how group-theoretic ideas arise across many areas of mathematics.
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 350 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: What is a Group?
Chapter 3: Important Families of Groups
Chapter 4: Lagrange’s Theorem and Cauchy’s Theorem
Chapter 5: Quotient Groups
Chapter 6: The Isomorphism Theorems
Chapter 7: The Structure Theorem for Finitely Generated Abelian Groups
Chapter 8: Divisible and Torsion Groups
Chapter 9: Groups Acting on Sets
Chapter 10: Burnside’s Counting Theorem
Chapter 11: The Sylow Theorems
Chapter 12: Geometric Group Actions
Chapter 13: Semidirect Products
Chapter 14: When is Every Group of Order n…?
Appendix A: Zorn’s Lemma and its Uses
Publication details
Publisher: CRC Press
Publication date: forthcoming
ISBN: to be announced
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