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Taking a slightly different approach from similar texts, **Introduction to Abstract Algebra** presents abstract algebra as the main tool underlying discrete mathematics and the digital world. It helps students fully understand groups, rings, semigroups, and monoids by rigorously building concepts from first principles.

*A Quick Introduction to Algebra*

The first three chapters of the book show how functional composition, cycle notation for permutations, and matrix notation for linear functions provide techniques for practical computation. The author also uses equivalence relations to introduce rational numbers and modular arithmetic as well as to present the first isomorphism theorem at the set level.

*The Basics of Abstract Algebra for a First-Semester Course*

Subsequent chapters cover orthogonal groups, stochastic matrices, Lagrange’s theorem, and groups of units of monoids. The text also deals with homomorphisms, which lead to Cayley’s theorem of reducing abstract groups to concrete groups of permutations. It then explores rings, integral domains, and fields.

*Advanced Topics for a Second-Semester Course*

The final, mostly self-contained chapters delve deeper into the theory of rings, fields, and groups. They discuss modules (such as vector spaces and abelian groups), group theory, and quasigroups.

**Numbers**

Ordering Numbers

The Well-Ordering Principle

Divisibility

The Division Algorithm

Greatest Common Divisors

The Euclidean Algorithm

Primes and Irreducibles

The Fundamental Theorem of Arithmetic **Functions **

Specifying Functions

Composite Functions

Linear Functions

Semigroups of Functions

Injectivity and Surjectivity

Isomorphisms

Groups of Permutations **Equivalence**

Kernel and Equivalence Relations

Equivalence Classes

Rational Numbers

The First Isomorphism Theorem for Sets

Modular Arithmetic **Groups and Monoids**

Semigroups

Monoids

Groups

Componentwise Structure

Powers

Submonoids and Subgroups

Cosets

Multiplication Tables **Homomorphisms**

Homomorphisms

Normal Subgroups

Quotients

The First Isomorphism Theorem for Groups

The Law of Exponents

Cayley’s Theorem **Rings**

Rings

Distributivity

Subrings

Ring Homomorphisms

Ideals

Quotient Rings

Polynomial Rings

Substitution **Fields**

Integral Domains

Degrees

Fields

Polynomials over Fields

Principal Ideal Domains

Irreducible Polynomials

Lagrange Interpolation

Fields of Fractions **Factorization**

Factorization in Integral Domains

Noetherian Domains

Unique Factorization Domains

Roots of Polynomials

Splitting Fields

Uniqueness of Splitting Fields

Structure of Finite Fields

Galois Fields **Modules**

Endomorphisms

Representing a Ring

Modules

Submodules

Direct Sums

Free Modules

Vector Spaces

Abelian Groups **Group Actions**

Actions

Orbits

Transitive Actions

Fixed Points

Faithful Actions

Cores

Alternating Groups

Sylow Theorems **Quasigroups**

Quasigroups

Latin Squares

Division

Quasigroup Homomorphisms

Quasigroup Homotopies

Principal Isotopy

Loops **Index** *Exercises, Study Projects, and Notes appear at the end of each chapter.*

… The author goes the extra mile to build algebraic concepts by confronting the pedagogic and logical sequence groups-first or rings-first dilemma … a perfect pure math precursor to Grillet and Knapp’s works. … The book’s well-thought out sequence supports a set of useful statements on how to use its 11 chapters in a course … The book is also outstanding for self-study. … I recommend this book as second to none on abstract algebra for its content, style, and expository efficiency.

—*Computing Reviews*, January 2011

… a careful treatment of the principal topics of abstract algebra … This is an attractive book which could be read by everybody because the author supposes not so much knowledge from the reader and gives all the necessary information to continue the reading from [one] chapter to the next. The approach used by the author to introduce modules and group actions is new and innovative. The book is well written … students and even experienced researchers may benefit strongly from this book. …

—IACR Book Reviews, October 2010

… This compact book covers topics one would expect to find in an abstract algebra text. … Smith’s approach is carefully implemented, and topics flow logically from one chapter to the next. The writing is careful and rigorous, yet accessible to hardworking students. The problems are collected at the end of each chapter in two sets, with one set made up of shorter exercises. … This is an ideal text for an abstract algebra course comprised of mathematics students or CS students who have either a strong minor or second major in mathematics. …

—*Computing Reviews*, December 2009

One can trace the author’s research interests to the border between algebra and category theory, which gives the textbook its unique flavour.

—*EMS Newsletter*, March 2009

The book is well written and flows well. Readers looking for an alternative approach to abstract algebra should consider this volume.

—J.R. Burke, Gonzaga University, *CHOICE*, July 2009, Vol. 46, No. 11

This book is well written, interesting to read, and the proofs and examples are clear and clean.

—David F. Anderson, *Mathematical Reviews*, 2009e