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Schaum’s Outline Theory and Problems of Abstract Algebra (Second Edition) by FRANK AYRES and LLOYD R. JAISINGH



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Schaum’s Outline Theory and Problems of Abstract Algebra (Second Edition) written by FRANK AYRES , Jr., Ph.D. and LLOYD R. JAISINGH, Professor of Mathematics, Morehead State University This book on algebraic systems is designed to be used either as a supplement to current texts or as a stand-alone text for a course in modern abstract algebra at the junior and/or senior levels. In addition, graduate students can use this book as a source for review. As such, this book is intended to provide a solid foundation for future study of a variety of systems rather than to be a study in depth of any one or more. The basic ingredients of algebraic systems–sets of elements, relations, operations, and mappings–are discussed in the first two chapters. The format established for this book is as follows: . a simple and concise presentation of each topic . a wide variety of familiar examples . proofs of most theorems included among the solved problems . a carefully selected set of supplementary exercises
In this upgrade, the text has made an effort to use standard notations for the set of natural numbers, the set of integers, the set of rational numbers, and the set of real numbers. In addition, definitions are highlighted rather than being embedded in the prose of the text. Also, a new chapter (Chapter 10) has been added to the text. It gives a very brief discussion of Sylow Theorems and the Galois group. The text starts with the Peano postulates for the natural numbers in Chapter 3, with the various number systems of elementary algebra being constructed and their salient properties discussed. This not only introduces the reader to a detailed and rigorous development of these number systems but also provides the reader with much needed practice for the reasoning behind the properties of the abstract systems which follow. The first abstract algebraic system – the Group – is considered in Chapter 9. Cosets of a subgroup, invariant subgroups, and their quotient groups are investigated as well. Chapter 9 ends with the Jordan–Ho¨ lder Theorem for finite groups. Rings, Integral Domains Division Rings, Fields are discussed in Chapters 11–12 while Polynomials over rings and fields are then considered in Chapter 13. Throughout these chapters, considerable attention is given to finite rings. Vector spaces are introduced in Chapter 14. The algebra of linear transformations on a vector space of finite dimension leads naturally to the algebra of matrices (Chapter 15). Matrices are then used to solve systems of linear equations and, thus provide simpler solutions to a number of problems connected to vector spaces. Matrix polynomials are discussed in Chapter 16 as an example of a non-commutative polynomial ring. The characteristic polynomial of a square matrix over a field is then defined. The characteristic roots and associated invariant vectors of real symmetric matrices are used to reduce the equations of conics and quadric surfaces to standard form. Linear algebras are formally defined in Chapter 17 and other examples briefly considered. In the final chapter (Chapter 18), Boolean algebras are introduced and important applications to simple electric circuits are discussed. The co-author wishes to thank the staff of the Schaum’s Outlines group, especially Barbara Gilson, Maureen Walker, and Andrew Litell, for all their support. In addition, the co-author wishes to thank the estate of Dr. Frank Ayres, Jr. for allowing me to help upgrade the original text.

Schaum’s Outline Theory and Problems of Abstract Algebra (Second Edition) written by FRANK AYRES , Jr., Ph.D. and LLOYD R. JAISINGH cover the following topics.

  • PART I SETS AND RELATIONS

  • 1. Sets
    Introduction
    1.1 Sets
    1.2 Equal Sets
    1.3 Subsets of a Set
    1.4 Universal Sets
    1.5 Intersection and Union of Sets
    1.6 Venn Diagrams
    1.7 Operations with Sets
    1.8 The Product Set
    1.9 Mappings
    1.10 One-to-One Mappings
    1.11 One-to-One Mapping of a Set onto Itself
    Solved Problems
    Supplementary Problems

  • 2. Relations and Operations
    Introduction
    2.1 Relations
    2.2 Properties of Binary Relations
    2.3 Equivalence Relations
    2.4 Equivalence Sets
    2.5 Ordering in Sets
    2.6 Operations
    2.7 Types of Binary Operations
    2.8 Well-Defined Operations
    2.9 Isomorphisms
    2.10 Permutations
    2.11 Transpositions
    2.12 Algebraic Systems
    Solved Problems
    Supplementary Problems

  • PART II NUMBER SYSTEMS

  • 3. The Natural Numbers
    Introduction
    3.1 The Peano Postulates
    3.2 Addition on N
    3.3 Multiplication on N
    3.4 Mathematical Induction
    3.5 The Order Relations
    3.6 Multiples and Powers
    3.7 Isomorphic Sets
    Solved Problems
    Supplementary Problems

  • 4. The Integers
    Introduction
    4.1 Binary Relation
    4.2 Addition and Multiplication on J
    4.3 The Positive Integers
    4.4 Zero and Negative Integers
    4.5 The Integers
    4.6 Order Relations
    4.7 Subtraction ‘‘’’
    4.8 Absolute Value jaj
    4.9 Addition and Multiplication on Z
    4.10 Other Properties of Integers
    Solved Problems
    Supplementary Problems

  • 5. Some Properties of Integers
    Introduction
    5.1 Divisors
    5.2 Primes
    5.3 Greatest Common Divisor
    5.4 Relatively Prime Integers
    5.5 Prime Factors
    5.6 Congruences
    5.7 The Algebra of Residue Classes
    5.8 Linear Congruences
    5.9 Positional Notation for Integers
    Solved Problems
    Supplementary Problems

  • 6. The Rational Numbers
    Introduction
    6.1 The Rational Numbers
    6.2 Addition and Multiplication
    6.3 Subtraction and Division
    6.4 Replacement
    6.5 Order Relations
    6.6 Reduction to Lowest Terms
    6.7 Decimal Representation
    Solved Problems
    Supplementary Problems

  • 7. The Real Numbers
    Introduction
    7.1 Dedekind Cuts
    7.2 Positive Cuts
    7.3 Multiplicative Inverses
    7.4 Additive Inverses
    7.5 Multiplication on K
    7.6 Subtraction and Division
    7.7 Order Relations
    7.8 Properties of the Real Numbers
    Solved Problems
    Supplementary Problems

  • 8. The Complex Numbers
    Introduction
    8.1 Addition and Multiplication on C
    8.2 Properties of Complex Numbers
    8.3 Subtraction and Division on C
    8.4 Trigonometric Representation
    8.5 Roots
    8.6 Primitive Roots of Unity
    Solved Problems
    Supplementary Problems

  • 9. Groups
    Introduction
    9.1 Groups
    9.2 Simple Properties of Groups
    9.3 Subgroups
    9.4 Cyclic Groups
    9.5 Permutation Groups
    9.6 Homomorphisms
    9.7 Isomorphisms
    9.8 Cosets
    9.9 Invariant Subgroups
    9.10 Quotient Groups
    9.11 Product of Subgroups
    9.12 Composition Series
    Solved Problems
    Supplementary Problems

  • 10. Further Topics on Group Theory
    Introduction
    10.1 Cauchy’s Theorem for Groups
    10.2 Groups of Order 2p and p2
    10.3 The Sylow Theorems
    10.4 Galois Group
    Solved Problems
    Supplementary Problems

  • 11. Rings
    Introduction
    11.1 Rings
    11.2 Properties of Rings
    11.3 Subrings
    11.4 Types of Rings
    11.5 Characteristic
    11.6 Divisors of Zero
    11.7 Homomorphisms and Isomorphisms
    11.8 Ideals
    11.9 Principal Ideals
    11.10 Prime and Maximal Ideals
    11.11 Quotient Rings
    11.12 Euclidean Rings
    Solved Problems
    Supplementary Problems

  • 12. Integral Domains, Division Rings, Fields
    Introduction
    12.1 Integral Domains
    12.2 Unit, Associate, Divisor
    12.3 Subdomains
    12.4 Ordered Integral Domains
    12.5 Division Algorithm
    12.6 Unique Factorization
    12.7 Division Rings
    12.8 Fields
    Solved Problems
    Supplementary Problems

  • 13. Polynomials
    Introduction
    13.1 Polynomial Forms
    13.2 Monic Polynomials
    13.3 Division
    13.4 Commutative Polynomial Rings with Unity
    13.5 Substitution Process
    13.6 The Polynomial Domain F½x
    13.7 Prime Polynomials
    13.8 The Polynomial Domain C½x
    13.9 Greatest Common Divisor
    13.10 Properties of the Polynomial Domain F½x
    Solved Problems
    Supplementary Problems

  • 14. Vector Spaces
    Introduction
    14.1 Vector Spaces
    14.2 Subspace of a Vector Space
    14.3 Linear Dependence
    14.4 Bases of a Vector Space
    14.5 Subspaces of a Vector Space
    14.6 Vector Spaces Over R
    14.7 Linear Transformations
    14.8 The Algebra of Linear Transformations
    Solved Problems
    Supplementary Problems

  • 15. Matrices
    Introduction
    15.1 Matrices
    15.2 Square Matrices
    15.3 Total Matrix Algebra
    15.4 A Matrix of Order m  n
    15.5 Solutions of a System of Linear Equations
    15.6 Elementary Transformations on a Matrix
    15.7 Upper Triangular, Lower Triangular, and Diagonal Matrices
    15.8 A Canonical Form
    15.9 Elementary Column Transformations
    15.10 Elementary Matrices
    15.11 Inverses of Elementary Matrices
    15.12 The Inverse of a Non-Singular Matrix
    15.13 Minimum Polynomial of a Square Matrix
    15.14 Systems of Linear Equations
    15.15 Systems of Non-Homogeneous Linear Equations
    15.16 Systems of Homogeneous Linear Equations
    15.17 Determinant of a Square Matrix
    15.18 Properties of Determinants
    15.19 Evaluation of Determinants
    Solved Problems
    Supplementary Problems

  • 16. Matrix Polynomials
    Introduction
    16.1 Matrices with Polynomial Elements
    16.2 Elementary Transformations
    16.3 Normal Form of a -Matrix
    16.4 Polynomials with Matrix Coefficients
    16.5 Division Algorithm
    16.6 The Characteristic Roots and Vectors of a Matrix
    16.7 Similar Matrices
    16.8 Real Symmetric Matrices
    16.9 Orthogonal Matrices
    16.10 Conics and Quadric Surfaces
    Solved Problems
    Supplementary Problems

  • 17. Linear Algebras
    Introduction
    17.1 Linear Algebra
    17.2 An Isomorphism
    Solved Problems
    Supplementary Problems

  • 18. Boolean Algebras
    Introduction
    18.1 Boolean Algebra
    18.2 Boolean Functions
    18.3 Normal Forms
    18.4 Changing the Form of a Boolean Function
    18.5 Order Relation in a Boolean Algebra
    18.6 Algebra of Electrical Networks
    18.7 Simplification of Networks
    Solved Problems
    Supplementary Problems

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