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Contemporary Abstract Algebra

Joseph A. Gallian
Houghton Mifflin
Publication Date: 
Number of Pages: 
[Reviewed by
Kara Shane Colley
, on

When reviewing a text, the question “How well will the students understand the subject just by reading this book?” always runs through the back of my mind. And according to this criterion, Contemporary Abstract Algebra by Joseph Gallian is a well-written book. After introducing a new concept or theorem, he always provides a plethora of examples. He explains, “The best way to grasp the meat of a theorem is to see what it says in specific cases.” Furthermore, Gallian keeps the big picture in mind. Rather than just proving theorem after theorem, he often steps back to explain why a theorem is important or how the theorem can be used, or even to explain what a theorem means in words, rather than just in symbols. Another plus are the numerous tables he includes, which sum up a lot of information in a concise way. For example, there is a summary of group examples and their properties.

What makes Contemporary Abstract Algebra unique is Gallian´s focus on showing that abstract algebra is a contemporary subject. He incorporates examples of physics, cryptography, chemistry, and computer science into the text. For instance, there is a description of how your credit card number is encrypted when buying online from In another section, he explains how molecules with chemical formulas of the form AB4, such as methane (CH4), have the same symmetry as the group A4. Gallian also shows that abstract algebra is an ever-expanding field of research by telling stories of how recent mathematicians pushed to solve certain problems. For example, he gives the history of “the enormous effort put forth by hundreds of mathematicians” since the 1960s “to discover and classify all finite simple groups.”

Contemporary Abstract Algebra is appropriate for a first or second course in abstract algebra. The text does not spend much time on preliminary number theory topics, like the division algorithm or modular arithmetic, so the students need to have familiarity with these topics. The text does provide a solid introduction to the traditional topics of groups, rings, and fields, but there is depth in his coverage of these topics. He includes, for example, internal and external direct products and the Fundamental Theorem of Finite Abelian Groups, in his section on groups. The special topics cover a selection of interesting themes, such as Cayley digraphs, which “provide a method of visualizing groups” and cyclotomic extensions which tie together many themes explored in the text.

The text also has some interesting extras. Gallian starts and ends each chapter with quotes from famous mathematicians, popular songs, and even few from Homer Simpson. Some of the quotes are simply amusing (“If you really want something in this life, you have to work for it — Now quiet, they’re about to announce the lottery numbers.” –Homer Simpson). Others offer sound mathematical advice (“‘For example’ is not proof.” –Jewish proverb). At the end of each chapter, he offers a list of suggested readings with summaries, as well as suggested websites, and films. The text also has a companion website which has true/false questions and software for computer exercises.

In the preface, Gallian lays out his goals for the text. Briefly, they are to give the student “a solid introduction to the traditional topics,” to show readers that “abstract algebra is a contemporary subject,” to provide students with an enjoyable text, and finally to help students gain competency in doing computations and writing proofs. He definitely meets all of these goals, and as such, I certainly recommend this textbook.

Kara Shane Colley studied physics at Dartmouth College and math education at Teachers College. She is currently taking a break from teaching math to volunteer at a meditation center in Mexico.


Note: Each chapter concludes with Exercises.

  • I. Integers and Equivalence Relations
  • Preliminaries
    Properties of Integers
    Modular Arithmetic
    Mathematical Induction
    Equivalence Relations
    Functions (Mappings)
    Computer Exercises
  • II. Groups
  • 1. Introduction to Groups
    Symmetries of a Square
    The Dihedral Groups
    Biography of Neils Abel
  • 2. Groups
    Definition and Examples of Groups
    Elementary Properties of Groups
    Historical Note
    Computer Exercises
  • 3. Finite Groups; Subgroups
    Terminology and Notation
    Subgroup Tests
    Examples of Subgroups
    Computer Exercises
  • 4. Cyclic Groups
    Properties of Cyclic Groups
    Classification of Subgroups of Cyclic Groups
    Computer Exercises
    Biography of J. J. Sylvester
  • Supplementary Exercises for Chapters 1-4
  • 5. Permutation Groups
    Definition and Notation
    Cycle Notation
    Properties of Permutations
    A Check-Digit Scheme Based on D5
    Computer Exercises
    Biography of Augustin Cauchy
  • 6. Isomorphisms
    Definition and Examples
    Cayley's Theorem
    Properties of Isomorphisms
    Biography of Arthur Cayley
  • 7. Cosets and Lagrange's Theorem
    Properties of Cosets
    Lagrange's Theorem and Consequences
    An Application of Cosets to Permutation Groups
    The Rotation Group of a Cube and a Soccer Ball
    Computer Exercises
    Biography of Joseph Lagrange
  • 8. External Direct Products
    Definition and Examples
    Properties of External Direct Products
    The Group of Units Modulo n as an External Direct Product
    Computer Exercises
    Biography of Leonard Adleman
  • Supplementary Exercises for Chapters 5-8
  • 9. Normal Subgroups and Factor Groups
    Normal Subgroups
    Factor Groups
    Applications of Factor Groups
    Internal Direct Products
    Biography of Évariste Galois
  • 10. Group Homomorphisms
    Definition and Examples
    Properties of Homomorphisms
    The First Isomorphism Theorem
    Computer Exercises
    Biography of Camille Jordan
  • 11. Fundamental Theorem of Finite Abelian Groups
    The Fundamental Theorem
    The Isomorphism Classes of Abelian Groups
    Proof of the Fundamental Theorem
    Computer Exercises
  • Supplementary Exercises for Chapters 9-11
  • III. Rings
  • 12. Introduction to Rings
    Motivation and Definition
    Examples of Rings
    Properties of Rings
    Computer Exercises
    Biography of I. N. Herstein
  • 13. Integral Domains
    Definition and Examples
    Characteristic of a Ring
    Computer Exercises
    Biography of Nathan Jacobson
  • 14. Ideals and Factor Rings
    Factor Rings
    Prime Ideals and Maximal Ideals
    Biography of Richard Dedekind
    Biography of Emmy Noether
  • Supplementary Exercises for Chapters 12-14
  • 15. Ring Homomorphisms
    Definition and Examples
    Properties of Ring Homomorphisms
    The Field of Quotients
  • 16. Polynomial Rings
    Notation and Terminology
    The Division Algorithm and Consequences
    Biography of Saunders Mac Lane
  • 17. Factorization of Polynomials
    Reducibility Tests
    Irreducibility Tests
    Unique Factorization in Z [x]
    Weird Dice: An Application of Unique Factorization
    Computer Exercises
  • 18. Divisibility in Integral Domains
    Irreducibles, Primes
    Historical Discussion of Fermat's Last Theorem
    Unique Factorization Domains
    Euclidean Domains
    Biography of Sophie Germain
    Biography of Andrew Wiles
  • Supplementary Exercises for Chapters 15-18
  • IV. Fields
  • 19. Vector Spaces
    Definition and Examples
    Linear Independence
    Biography of Emil Artin
    Biography of Olga Taussky-Todd
  • 20. Extension Fields
    The Fundamental Theorem of Field Theory
    Splitting Fields
    Zeros of an Irreducible Polynomial
    Biography of Leopold Kronecker
  • 21. Algebraic Extensions
    Characterization of Extensions
    Finite Extensions
    Properties of Algebraic Extensions
    Biography of Irving Kaplansky
  • 22. Finite Fields
    Classification of Finite Fields
    Structure of Finite Fields
    Subfields of a Finite Field
    Computer Exercises
    Biography of L. E. Dickson
  • 23. Geometric Constructions
    Historical Discussion of Geometric Constructions
    Constructible Numbers
    Angle-Trisectors and Circle-Squarers
  • Supplementary Exercises for Chapters 19-23
  • V. Special Topics
  • 24. Sylow Theorems
    Conjugacy Classes
    The Class Equation
    The Probability That Two Elements Commute
    The Sylow Theorems
    Applications of Sylow Theorems
    Biography of Ludvig Sylow
  • 25. Finite Simple Groups
    Historical Background
    Nonsimplicity Tests
    The Simplicity of A5
    The Fields Medal
    The Cole Prize
    Computer Exercises
    Biography of Michael Aschbacher
    Biography of Daniel Gorenstein
    Biography of John Thompson
  • 26. Generators and Relations
    Definitions and Notation
    Free Group
    Generators and Relations
    Classification of Groups of Order up to 15
    Characterization of Dihedral Groups
    Realizing the Dihedral Groups with Mirrors
    Biography of Marshall Hall, Jr.
  • 27. Symmetry Groups
    Classification of Finite Plane Symmetry Groups
    Classification of Finite Group of Rotations in R3
  • 28. Frieze Groups and Crystallographic Groups
    The Frieze Groups
    The Crystallographic Groups
    Identification of Plane Periodic Patterns
    Biography of M. C. Escher
    Biography of George Pólya
    Biography of John H. Conway
  • 29. Symmetry and Counting
    Burnside's Theorem
    Group Action
    Biography of William Burnside
  • 30. Cayley Digraphs of Groups
    The Cayley Digraph of a Group
    Hamiltonian Circuits and Paths
    Some Applications
    Biography of William Rowan Hamilton
    Biography of Paul Erdös
  • 31. Introduction to Algebraic Coding Theory
    Linear Codes
    Parity-Check Matrix Decoding
    Coset Decoding
    Historical Note: Reed-Solomon Codes
    Biography of Richard W. Hamming
    Biography of Jessie MacWilliams
    Biography of Vera Pless
  • 32. An Introduction to Galois Theory
    Fundamental Theorem of Galois Theory
    Solvability of Polynomials by Radicals
    Insolvability of a Quintic
    Biography of Philip Hall
  • 33. Cyclotomic Extensions
    Cyclotomic Polynomials
    The Constructible Regular n-gons
    Computer Exercise
    Biography of Carl Friedrich Gauss
    Biography of Manjul Bhargava
  • Supplementary Exercises for Chapters 24-33