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Group theory is the branch of mathematics that studies symmetry, found in crystals, art, architecture, music and many other contexts. But its beauty is lost on students when it is taught in a technical style that is difficult to understand. Visual Group Theory assumes only a high school mathematics background and covers a typical undergraduate course in group theory from a thoroughly visual perspective.
The more than 300 illustrations in Visual Group Theory bring groups, subgroups, homomorphisms, products, and quotients into clear view. Every topic and theorem is accompanied with a visual demonstration of its meaning and import, from the basics of groups and subgroups through advanced structural concepts such as semidirect products and Sylow theory.
Acknowledgments
Preface
Overview
What is a group?
A famous toy
Considering the cube
The study of symmetry
Rules of a group
Exercises
What do groups look like?
Mapmaking
A not-so-famous toy
Mapping a group
Cayley diagrams
A touch more abstract
Exercises
Why study groups?
Groups of symmetries
Groups of actions
Groups everywhere
Exercises
Algebra at last
Where have all the actions gone?
Combine, combine, combine
Multiplication tables
The classic definition
Exercises
Five families
Cyclic groups
Abelian groups
Dihedral groups
Symmetric and alternating groups
Exercises
Subgroups
What multiplication tables say about Cayley diagrams
Seeing subgroups
Revealing subgroups
Cosets
Lagrange's theorem
Exercises
Products and quotients
The direct product
Semidirect products
Normal subgroups and quotients
Normalizers
Conjugacy
Exercises
The power of homomorphisms
Embeddings and quotient maps
The Fundamental Homomorphism Theorem
Modular arithmetic
Direct products and relatively prime numbers
The Fundamental Theorem of Abelian Groups
Semidirect products revisited
Exercises
Sylow theory
Group actions
Approaching Sylow: Cauchy's Theorem
p-groups
Sylow Theorems
Exercises
Galois theory
The big question
More big questions
Visualizing field extensions
Irreducible polynomials
Galois groups
The heart of Galois theory
Unsolvability
Exercises
Answers to selected Exercises
Bibliography
Index of Symbols Used
Index
About the Author