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Modern Algebra: An Introduction

John R. Durbin
John Wiley
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Russell Jay Hendel
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This book is intended as an undergraduate text for abstract algebra. It covers the standard material and standard applications sufficient for a two-semester sequence in abstract algebra: see the table of contents.

The book is excellent, especially for students planning to go into fields with applications of abstract algebra, for example chemistry majors and students interested in coding theory. The book is also very useful for undergraduates with no specific applications in mind.

The strongest attributes of the book are a) its conciseness b) the richness of applications, c) the hands-on computational approach of the book, d) the diversity of problems, and e) the superb bibliography and supplemental readings.

Conciseness: The book has 71 sections (66 in the text and 5 more on the website). Each section is sufficiently rich to take one class lecture. Furthermore, despite the book’s conciseness, the examples in each section have a rich diversity. Section 5, for example, introduces groups, then gives the following ten examples of groups: even integers, positive integers, the singleton set {0} under addition, the rationals, an abstract group (presented by a Cayley table) of order 3, the set of all invertible mappings of a set under composition, the set of rotations about a point, linear mappings, the 2×2 real matrices under addition, and the set of 2×2 real matrices with non-zero determinants under multiplication. Consequently, a student just introduced to groups is exposed to sufficient examples to appreciate the full generality of the group concept.

Applications: Despite the conciseness of the book, all the standard applications are treated. There are also several applications not found in many textbooks. Besides the standard topics of codes, Boolean switching theory, Galois Theory, geometric constructions, and solvability, there are delightful sections on symmetry, crystallographic groups and the Burnside counting theorem.

Problems: As an instructor, I don’t always learn from a textbook. Here are just two exercises from this book which I could not just sit down and solve:

Exercise 57.3 Consider the problem of painting the edges of a square so that one is red, one is white, one is blue, and one is yellow. (a) In how many distinguishable ways can this be done if the edges of the square are distinguishable? (b) Repeat (a) except count different ways as being indistinguishable if one can be obtained from the other by rotation of the square in the plane. (c) repeat (b), except permit reflections through lines as well as rotations in the plane.

Exercise 42.17 (d) Carry out the process outlined here to solve by radicals x3 + 4x2 + 4x + 3 = 0.

Computational Approach: Abstract algebra is challenging to many students because of its abstractness. It is easy for the student to get lost. Durbin (partially) remedies this problem by presenting many hand-on aids: Group laws are supplemented with Cayley tables, permutations are presented with the standard “easy to read” two-row form for permutations, groups based on geometric transformations are supplemented with clear pictures and diagrams. This is a tremendous aid to the weaker student. Frequently, when I teach abstract algebra it is either my job or the student’s to provide these aids. It is refreshing to see the textbook itself provide them. Furthermore, despite the book’s conciseness, the examples are non-trivial. One quick illustration: Tables 50.3 and 50.4 present the Cayley tables for the field Z3[X]/(1+x3), a field of order 9.

Problem Diversity: These days it is standard for “good” algebra textbooks to have problems with different difficulty levels. In this book there are two levels — standard problems reviewing section material and more advanced problems (including those encouraging proofs). The standard even-odd division of problems is used with the solutions to odd numbered problems at the end of the book. The website has more material including supplementary problems, solution manuals etc.

Bibliography: Refreshingly, each chapter ends with a section entitled “Notes on Chapter...” These notes are really extensive bibliographies: Other texts and websites with information on the chapter content. The bibliography is diverse - both technical as well as popular books are mentioned. I almost overlooked this wonderful feature of the book - I would recommend the author call this section “Bibliography” or “Supplemental Reading” in future additions.

My only minor criticism of the book is the carve-out of five sections on coding theory and Boolean switching theory to the website. I would prefer these sections be retained in the book proper. The sections on the website are, however, complete with many examples, challenging topics, and non-trivial exercises.

The location of the website could have been made clearer. The author tells the instructor that he may find the material on codes and switching theory at A little more information should have been supplied. One has to a) click on the “Modern Algebra” book icon, b) click on “Companion Sites” under the “Instructors” grouping, c) click on “Instructor Companion Site”, and then d) Click on “Web only material”. One must also know that Appendix F contains four very thorough sections on coding theory (including linear codes, using finite fields, and computing error rates) while Appendix G contains a standard section on Boolean switching theory. Links to an “errata” sheet and an “Instructor Solution Manual” may also be found on this page.

See also our review of the fifth edition.



Russell Jay Hendel holds a Ph.D. in theoretical mathematics and an Associateship from the Society of Actuaries. He teaches at Towson University. His interests include discrete number theory, applications of technology to education, problem writing, actuarial science and the interaction between mathematics, art and poetry.


I.  Mappings and Operations
1 Mappings
2 Composition
4 Composition as an Operation

II.  Introduction to Groups
5 Definition and Examples
6 Permutations
7 Subgroups
8 Groups and Symmetry

III.  Equivalence. Congruence. Divisibility
9 Equivalence Relations
10 Congruence.  The Division Algorithm
11 Integers Modulo n
12 Greatest Common Divisors.  The Euclidean Algorithm
13 Factorization.  Euler's Phi-Function

IV.  Groups
14 Elementary Properties
15  Generators.  Direct Products
16 Cosets
17 Lagrange's Theorem.  Cyclic Groups
18 Isomorphism
19 More on Isomorphism
20 Cayley's Theorem
Appendix: RSA Algorithm

V.  Group Homomorphisms
21 Homomorphisms of Groups.  Kernels
22 Quotient Groups
23 The Fundamental Homomorphism Theorm

VI.  Introduction to Rings
24 Definition and Examples
25 Integral Domains.  Subrings
26 Fields
27 Isomorphism.  Characteristic

VII.  The Familiar Number Systems
28 Ordered Integral Domains
29 The Integers
30 Field of Quotients.  The Field of Rational Numbers
31 Ordered Fields.  The Field of Real Numbers
32 The Field of Complex Numbers
33 Complex Roots of Unity

VIII.  Polynomials
34 Definition and Elementary Properties
35 The Division Algorithm
36 Factorization of Polynomials
37 Unique Factorization Domains

IX.  Quotient Rings
38 Homomorphisms of Rings. Ideals
39 Quotient Rings
40 Quotient Rings of F[X]
41 Factorization and Ideals

X.  Galois Theory: Overview
42 Simple Extensions.  Degree
43 Roots of Polynomials
44 Fundamental Theorem: Introduction

XI.  Galois Theory
45 Galois Groups
46 Splitting Fields.  Galois Groups
47 Separability and Normality
48 Fundamental Theorem of Galois Theory
49 Solvability by Radicals
50 Finite Fields

XII.  Geometric Constructions
51 Three Famous Problems
52 Constructible Numbers
53 Impossible Constructions

XIII.  Solvable and Alternating Groups
54 Isomorphis Theorems.  Solvable Groups
55 Alternation Groups

XIV.  Applications of Permutation Groups
56 Groups Acting on Sets
57 Burnside's Counting Theorem
58 Sylow's Theorem

XV.  Symmetry
59 Finite Symmetry Groups
60 Infinite Two-dimensional Symmetry Groups
61 On Crystallographic Groups
62 The Euclidean Group

XVI.  Lattices and Boolean Algebras
63 Partially Ordered Sets
64 Lattices
65 Boolean Algebras
66 Finite Boolean Algebras

A.  Sets
B.  Proofs
C.  Mathematical Induction
D.  Linear Algebra
E.  Solutions to Selected Problems
Photo Credit List
Index of Notation