An Introduction to Algebraic Structures

Overview


As the author notes in the preface, "The purpose of this book is to acquaint a broad spectrum of students with what is today known as 'abstract algebra.'" Written for a one-semester course, this self-contained text includes numerous examples designed to base the definitions and theorems on experience, to illustrate the theory with concrete examples in familiar contexts, and to give the student extensive computational practice.
The first three chapters progress in a relatively ...
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An Introduction to Algebraic Structures

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Overview


As the author notes in the preface, "The purpose of this book is to acquaint a broad spectrum of students with what is today known as 'abstract algebra.'" Written for a one-semester course, this self-contained text includes numerous examples designed to base the definitions and theorems on experience, to illustrate the theory with concrete examples in familiar contexts, and to give the student extensive computational practice.
The first three chapters progress in a relatively leisurely fashion and include abundant detail to make them as comprehensible as possible. Chapter One provides a short course in sets and numbers for students lacking those prerequisites, rendering the book largely self-contained. While Chapters Four and Five are more challenging, they are well within the reach of the serious student.
The exercises have been carefully chosen for maximum usefulness. Some are formal and manipulative, illustrating the theory and helping to develop computational skills. Others constitute an integral part of the theory, by asking the student to supply proofs or parts of proofs omitted from the text. Still others stretch mathematical imaginations by calling for both conjectures and proofs.
Taken together, text and exercises comprise an excellent introduction to the power and elegance of abstract algebra. Now available in this inexpensive edition, the book is accessible to a wide range of students, who will find it an exceptionally valuable resource.
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Product Details

  • ISBN-13: 9780486659404
  • Publisher: Dover Publications
  • Publication date: 10/18/2010
  • Series: Dover Books on Mathematics Series
  • Edition description: Unabridged
  • Pages: 272
  • Product dimensions: 5.42 (w) x 8.48 (h) x 0.53 (d)

Meet the Author

A Professor Emeritus at the University of Illinois, Joseph Landin served as Head of the Department of Mathematics for 10 years.

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Table of Contents

1. Sets and Numbers
  I. THE ELEMENTS OF SET THEORY
    1. The Concept of Set
    2. "Constants, Variables and Related Matters"
    3. Subsets and Equality of Sets
    4. The Algebra of Sets; The Empty Set
    5. A Notation for Sets
    6. Generalized Intersection and Union
    7. Ordered Pairs and Cartesian Products
    8. Functions (or Mappings)
    9. A Classification of Mappings
    10. Composition of Mappings
    11. Equivalence Relations and Partititions
  II. THE REAL NUMBERS
    12. Introduction
    13. The Real Numbers
    14. The Natural Numbers
    15. The Integers
    16. The Rational Numbers
    17. The Complex Numbers
2. The Theory of Groups
  1. The Group Concept
  2. Some Simple Consequences of the Definition of Group
  3. Powers of Elements in a Group
  4. Order of a Group; Order of a Group Element
  5. Cyclic Groups
  6. The Symmetric Groups
  7. Cycles; Decomposition of Permutations into Disjoint Cycles
  8. Full Transformation Groups
  9. Restrictions of Binary Operations
  10. Subgroups
  11. A Discussion of Subgroups
  12. The Alternating Group
  13. The Congruence of Integers
  14. The Modular Arithmetics
  15. Equivalence Relations and Subgroups
  16. Index of a Subgroup
  17. "Stable Relations, Normal Subgroups, Quotient Groups"
  18. Conclusion
3. Group Isomorphism and Homomorphism
  1. Introduction
  2. "Group Isomorphism; Examples, Definitions and Simplest Properties"
  3. The Isomorphism Theorem for the Symmetric Groups
  4. The Theorem of Cayley
  5. Group Homomorphisms
  6. A Relation Between Epimorphisms and Isomorphisms
  7. Endomorphisms of a Group
4. The Theory of Rings
  1. Introduction
  2. Definition of Ring
  3. Some Properties of Rings
  4. "The Modular Arithmetics, Again"
  5. Integral Domains
  6. Fields
  7. Subrings
  8. Ring Homomorphisms
  9. Ideals
  10. Residue Class Rings
  11. Some Basic Homomorphism Theorems
  12. Principle Ideal and Unique Factorization Domains
  13. Prime and Maximal Ideals
  14. The Quotient Field of an Integral Domain
5. Polynomial Rings
  1. Introduction; The Concept of Polynomial Ring
  2. Indeterminates
  3. Existence of Indeterminates
  4. Polynomial Domains Over a Field
  5. Unique Factorization in Polynomial Domains
  6. Polynomial Rings in Two Indeterminates
  7. Polynomial Functions and Polynomials
  8. Some Characterizations of Intermediates
  9. Substitution Homomorphisms
  10. Roots of Polynomials
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