Visual Group Theory

Citation

Carter, N. (2009). Visual Group Theory. Mathematical Association of America (Classroom Resource Materials, vol. 32). Reissued by the American Mathematical Society, 2021.

Chapter Summary

The book runs to about 300 pages in ten chapters, preceded by a short overview and followed by answers to selected exercises. The author counts more than 300 images, more than one per page on average. Cayley diagrams are the main tool, supported by multiplication tables, objects with symmetry, cycle graphs, Hasse diagrams, action diagrams and homomorphism diagrams. Every chapter ends with exercises, and the overview notes that each chapter's exercises include some on infinite groups.

Chapter 1: What is a group?

Starting from a famous toy, the chapter considers a cube puzzle, introduces the study of symmetry and sets out the rules of a group. Groups are defined here as collections of actions, not yet as sets with an operation.

Chapter 2: What do groups look like?

Through mapmaking, a second, less famous toy, and the mapping of a group, the chapter arrives at Cayley diagrams, the book's central visual device. A final section makes the idea "a touch more abstract".

Chapter 3: Why study groups?

Motivation through groups of symmetries, groups of actions and examples of groups found in many settings ("Groups everywhere").

Chapter 4: Algebra at last

The algebra appears only here. After asking where the actions have gone, the chapter treats combining elements and multiplication tables, and reaches the classic definition of a group as a set with a binary operation.

Chapter 5: Five families

A survey of cyclic, abelian, dihedral, symmetric and alternating groups. The author describes this chapter as broad exposure to finite groups that is helpful but not strictly needed later, apart from the definition of abelian in Section 5.2.

Chapter 6: Subgroups

The chapter links multiplication tables to Cayley diagrams, then shows how subgroups can be seen and revealed in diagrams. It continues to cosets and Lagrange's theorem.

Chapter 7: Products and quotients

Direct products and semidirect products, normal subgroups and quotient groups, normalizers and conjugacy.

Chapter 8: The power of homomorphisms

Embeddings and quotient maps lead to the Fundamental Homomorphism Theorem. The chapter applies it to modular arithmetic, to direct products and relatively prime numbers, and to the Fundamental Theorem of Abelian Groups, then revisits semidirect products.

Chapter 9: Sylow theory

Group actions, Cauchy's Theorem as an approach to Sylow, p-groups and the Sylow Theorems.

Chapter 10: Galois theory

Starting from "the big question" of solving polynomial equations, the chapter visualizes field extensions, discusses irreducible polynomials and Galois groups, and reaches "the heart of Galois theory". It ends with unsolvability; the overview names the unsolvability of the quintic as the chapter's focus and endpoint. It includes an introduction to fields but states several theorems without proof.

Key Concepts

Chapters 1–4: Groups as actions, then as algebra

• Group as a collection of actions: the book's starting definition, built from puzzles and symmetries.
• Cayley diagram: a picture of a group in which nodes are elements and arrows show the effect of generators.
• Multiplication table: the operation table of a finite group, connected to Cayley diagrams.
• Classic definition: a group as a set with a binary operation satisfying the group axioms, reached in Chapter 4.

Chapters 5–6: Families and subgroups

• Five families: cyclic, abelian, dihedral, symmetric and alternating groups.
• Cosets and Lagrange's theorem: how a subgroup partitions a group, and why its order divides the group's order.

Chapters 7–8: Products, quotients and homomorphisms

• Direct and semidirect products: ways of building larger groups from smaller ones.
• Normal subgroups and quotients: when collapsing cosets produces a group.
• Fundamental Homomorphism Theorem: the relationship between a homomorphism, its kernel and its image.
• Fundamental Theorem of Abelian Groups: the structure theorem for (finite) abelian groups.

Chapters 9–10: Sylow theory and Galois theory

• Group actions and Cauchy's Theorem: the tools used to approach the Sylow Theorems.
• Sylow Theorems: results on subgroups of prime-power order.
• Galois group: the group of symmetries of a field extension, which connects polynomial equations to group structure.
• Unsolvability of the quintic: the result that closes the book.

Critical Analysis

Strengths

• Visual from the first page. With over 300 images, the book depends on pictures in a way few algebra texts do. The author explains why Cayley diagrams lead: they show group structure clearly and faithfully.
• A deliberate reversal of the usual order. Most texts define a group as a set with an operation and prove Cayley's Theorem later. Carter defines groups as collections of actions in Chapter 1 and reaches the classic definition in Chapter 4. Students who find axioms unmotivated meet the examples first.
• Covers a typical course. The publisher describes the book as covering a typical undergraduate group theory course. The contents bear this out, from subgroups and Lagrange's theorem through quotients, homomorphisms and the Fundamental Theorem of Abelian Groups to Sylow theory, with a chapter on Galois theory added. The Mathematical Association of America awarded it the Beckenbach Book Prize in 2012.
• Software companion. Many illustrations were made with Group Explorer, the author's free software, and its investigations can help with some exercises. Few exercises require it.

Limitations

• Not comprehensive, and light on proof practice. The preface says the book aims to teach readers to "know" groups rather than to cover group theory fully. Traditional texts train students to make and prove conjectures. Students in a proof-based course will need a conventional text alongside it.
• Groups only. Rings, fields and other structures are outside its scope apart from the introduction to fields in Chapter 10, where several theorems are stated without proof.
• Mostly finite groups. The author focuses on finite groups because they are easier to diagram and build intuition. The most common infinite groups appear, but readers interested mainly in infinite groups will find less.
• The software has moved. The preface gives a SourceForge address for Group Explorer; the current version (3.0) is a web application maintained on the author's GitHub pages.

Where It Fits in the Course

Carter's book is the visual entry point to this course's group theory. It suits students who have not yet taken abstract algebra, or who are in a first course and struggling to picture the definitions. Charles Pinter's A Book of Abstract Algebra and Thomas Judson's Abstract Algebra: Theory and Applications are full first-course textbooks that cover rings and fields as well as groups, in the conventional definition–theorem–proof style. Carter can be read before either, or chapter by chapter alongside their group theory chapters.

Joseph Gallian's Contemporary Abstract Algebra and Peter Cameron's Introduction to Algebra are fuller course texts in which the images from Carter can help with cosets, quotients and homomorphisms. R. B. J. T. Allenby's Rings, Fields and Groups is a more demanding treatment for later. None of these replaces Carter's pictures, and Carter does not replace their proofs and exercises. The most effective use is to pair them.

How to Use This Book

The preface addresses students beginning a first course in group theory, either in place of a textbook or as a supplement, as well as recreational readers who want an overview. Only high-school mathematics is assumed, together with a willingness to think analytically.

• Read in order. The overview says the structure is linear and later sections usually depend on earlier ones. There are two exceptions. Most of Chapter 5 can be skipped and returned to later, apart from the definition of abelian in Section 5.2. Chapter 10 depends on Chapter 9 only through Cauchy's Theorem (Section 9.2), which is used in Section 10.7.
• Core path for a group theory course: Chapters 1–4, then Chapters 6–8, which cover subgroups and cosets, products and quotients, and homomorphisms. Chapter 9 is the core for Sylow theory. Chapter 10 is an extension showing what group theory can do.
• Draw the pictures yourself. Redraw the Cayley diagrams, and use Group Explorer, the free web application, to experiment with the groups in each chapter.
• Do the exercises, then check selected answers in the appendix. Then turn to a conventional text for proof-writing practice on the same topics.