Contemporary Abstract Algebra

Citation

Gallian, J. A. (2025). Contemporary Abstract Algebra (11th ed.). Chapman & Hall/CRC.

Chapter Summary

The eleventh edition has a preliminary Chapter 0 and 31 numbered chapters: groups (1–11), rings and polynomials (12–18), fields (19–22) and a run of further topics (23–31). Nearly every chapter ends with exercises followed by one or more short biographies, and about 40 pages of selected answers close the book.

Chapter 0: Preliminaries

A review of the background the course needs: properties of the integers, modular arithmetic, complex numbers, mathematical induction, equivalence relations and functions.

Chapters 1–4: Introduction to Groups; Groups; Finite Groups; Subgroups; Cyclic Groups

Groups are introduced through the symmetries of a square and the dihedral groups before the formal definition, elementary properties and a historical note. The following chapters cover terminology, subgroup tests and examples of subgroups, then the properties of cyclic groups and the classification of their subgroups.

Chapters 5–8: Permutation Groups; Isomorphisms; Cosets and Lagrange's Theorem; External Direct Products

Permutation groups are followed by isomorphisms, automorphisms and Cayley's Theorem. The cosets chapter proves Lagrange's Theorem and applies cosets to permutation groups, to the rotation groups of a cube and a soccer ball, and to the Rubik's Cube; external direct products come with a section of applications.

Chapters 9–11: Normal Subgroups and Factor Groups; Group Homomorphisms; Fundamental Theorem of Finite Abelian Groups

The structural core of group theory: normal subgroups, factor groups and their applications, internal direct products, homomorphisms and the First Isomorphism Theorem, and the classification of finite abelian groups with a proof of the Fundamental Theorem.

Chapters 12–15: Introduction to Rings; Integral Domains; Ideals and Factor Rings; Ring Homomorphisms

Rings are motivated and defined, with examples, properties and subrings. Integral domains, fields and the characteristic of a ring follow, then factor rings, prime and maximal ideals, ring homomorphisms and the field of quotients.

Chapters 16–18: Polynomial Rings; Factorization of Polynomials; Divisibility in Integral Domains

Polynomial rings and the division algorithm lead to the factorization of polynomials, then to irreducibles and primes, unique factorization domains and Euclidean domains, with a historical discussion of Fermat's Last Theorem.

Chapters 19–22: Extension Fields; Algebraic Extensions; Finite Fields; Geometric Constructions

Field theory begins with the Fundamental Theorem of Field Theory, splitting fields and zeros of irreducible polynomials, continues with finite and algebraic extensions and the classification and structure of finite fields, and applies the theory to classical geometric constructions through constructible numbers.

Chapters 23–25: Sylow Theorems; Finite Simple Groups; Generators and Relations

A return to group theory: conjugacy classes, the class equation, the Sylow Theorems and their applications; finite simple groups; and presentations by generators and relations, including a classification of groups of order up to 15 and a characterization of dihedral groups.

Chapters 26–28: Symmetry Groups; Symmetry and Counting; Cayley Digraphs of Groups

Isometries and the classification of finite plane symmetry groups and finite rotation groups in three dimensions; Burnside's Theorem and group actions applied to counting problems; and Cayley digraphs as pictures of groups.

Chapter 29: Introduction to Algebraic Coding Theory

Linear codes, parity-check matrix decoding and coset decoding, with a historical note.

Chapters 30–31: An Introduction to Galois Theory; Cyclotomic Extensions

Two short closing chapters (about 14 and 10 pages) introduce Galois theory and cyclotomic extensions, drawing on the field theory of Chapters 19–21.

Key Concepts

Chapters 0–11: Groups

• Dihedral group: the group of symmetries of a regular polygon, the book's first example of a group.
• Lagrange's Theorem: the order of a subgroup of a finite group divides the order of the group.
• Factor group: the group formed by the cosets of a normal subgroup.
• First Isomorphism Theorem: the image of a group homomorphism is isomorphic to the factor group by its kernel.
• Fundamental Theorem of Finite Abelian Groups: every finite abelian group is a direct product of cyclic groups of prime-power order.

Chapters 12–18: Rings and Polynomials

• Integral domain: a commutative ring with unity and no zero-divisors.
• Prime and maximal ideals: the ideals whose factor rings are integral domains and fields respectively.
• Unique factorization domain: an integral domain in which every nonzero non-unit factors into irreducibles uniquely up to order and units.

Chapters 19–22: Fields

• Splitting field: the smallest extension field in which a given polynomial factors completely into linear factors.
• Finite field: a field with finitely many elements; such fields are classified by their order, a power of a prime.
• Constructible number: a length obtainable by straightedge and compass, the tool for proving classical constructions impossible.

Chapters 23–31: Special Topics

• Sylow Theorems: results guaranteeing the existence of subgroups of prime-power order and describing how many there are.
• Burnside's Theorem: a formula for counting orbits of a group action, used to count objects up to symmetry.
• Linear code: an error-correcting code that is a subspace, decoded with a parity-check matrix or cosets.

Critical Analysis

Strengths

• Examples and exercises in volume. The publisher lists about 1,900 computational and theoretical exercises, over 300 worked examples, and skeleton solutions and hints for the odd-numbered exercises at the back of the book (the selected answers run to about 40 pages). It also highlights exercises designed to build the habit of generalizing. Reviewing the second edition for zbMATH, T. S. Blyth already praised the clear exposition and the large number of exercises.
• Concrete applications built into the theory. The section headings show applications placed where the theory arises: cosets applied to the rotations of a cube and a soccer ball and to the Rubik's Cube, Burnside's Theorem applied to counting, a full chapter on algebraic coding theory, and Cayley digraphs. The publisher describes the applications as drawn from science, computing and everyday life.
• History and people. Nearly every chapter ends with biographies, from Abel, Cauchy and Galois to Alan Turing, Jessie MacWilliams, Vera Pless and Manjul Bhargava, and there are historical notes on topics such as Fermat's Last Theorem and geometric constructions. Blyth called the biographies a delight.
• Support beyond the printed text. The book links to interactive True/False questions and computer exercises on the author's website, which also offers advice for students learning proofs and abstract algebra. A Student's Solutions Manual covers all odd-numbered and many even-numbered exercises.

Limitations

• More than one course can use. With 31 chapters after the preliminaries, the book covers more than a typical one-semester course; instructors and self-learners must choose, and the special topics in Chapters 23–31 compete for limited time.
• Galois theory only introduced. The final two chapters on Galois theory and cyclotomic extensions are short, so students who need a full treatment of Galois theory will need a more advanced text afterwards.
• Some topics moved online. The print edition no longer contains a chapter on vector spaces or on frieze and crystallographic groups; the author's website posts those chapters from the ninth edition.
• Odd-numbered answers and a light tone. Blyth's two reservations about the second edition, answers given only to odd-numbered exercises and an excess of witty quotations, still describe the current book: the publisher advertises odd-numbered skeleton solutions and "motivational and humorous quotations".

Where It Fits in the Course

Gallian is a full-length standard textbook for the subject and a natural main text for this course. Among the gentler books in the list, Thomas Judson's Abstract Algebra: Theory and Applications is described as a free, openly licensed text covering groups, rings and fields with applications and optional Sage exercises; it is a close alternative to Gallian for a first course and a useful second explanation alongside it. Nathan Carter's Visual Group Theory builds intuition about groups through Cayley diagrams and free software before formal algebra, and makes a good companion to Chapters 1–11; Gallian's website links to Group Explorer, the software used with Carter's book.

Peter Cameron's Introduction to Algebra is described as starting from proofs, sets and relations and covering rings, groups, vector spaces and modules, so it supplies the linear algebra that Gallian's print edition leaves out. Charles Pinter's A Book of Abstract Algebra is a gentle first course in the subject and works as an accessible alternative or companion to Gallian in a first course. R. B. J. T. Allenby's Rings, Fields and Groups is an introduction to abstract algebra that treats rings and fields before groups, the reverse of Gallian's order, and can give students a second view of the same structures.

How to Use This Book

The publisher describes the book as intended for a course whose main purpose is to enable students to do computations and write proofs, so it suits mathematics majors and prospective teachers after a first proof-based course, as well as computer science and physics students who need group and ring theory.

A practical plan:

• Start with Chapter 0 only as far as needed; modular arithmetic, equivalence relations and induction are the tools the early group chapters draw on.
• First semester (groups): Chapters 1–11, which take the reader from symmetries of a square to the classification of finite abelian groups. Chapter 23 (Sylow Theorems) can be added at the end for a course that stays with groups.
• Second semester (rings and fields): Chapters 12–22, then a selection from the special topics. Chapters 30–31 on Galois theory depend on the field theory of Chapters 19–21.
• Special topics for projects: Chapters 26–29 (symmetry, counting, Cayley digraphs, coding theory) are well suited to student projects or presentations.

Work the examples before attempting the exercises, and do a mix of computational and proof exercises in each chapter. Use the True/False questions on the author's website to test understanding after reading each chapter, try the computer exercises to make and test conjectures, and use the selected answers or the Student's Solutions Manual to check work on odd-numbered exercises.