Book:Thomas A. Whitelaw/An Introduction to Abstract Algebra

Subject Matter

 * Logic
 * Set Theory
 * Mappngs
 * Group Theory
 * Ring Theory

Contents

 * Sets and Logic
 * Some very general remarks
 * Introductory remarks on sets
 * Statements and conditions; quantifiers
 * The implies sign
 * Proof by contradiction
 * Subsets
 * Unions and intersections
 * Cartesian product of sets


 * Some Properties of $\Z$
 * Introduction
 * The well-ordering principle
 * The division algorithm
 * Highest common factors and Euclid's algorithm
 * The fundamental theorem of arithmetic
 * Congruence modulo $m$ ($$m \in \N$$)


 * Equivalence Relations and Equivalence Classes
 * Relations in general
 * Equivalence relations
 * Equivalence classes
 * Congruence classes
 * Properties of $$\Z_m$$ as an algebraic system


 * Mappings
 * Introduction
 * The image of a subset of the domain; surjections
 * Injections; bijections; inverse of a bijection
 * Restriction of a Mapping
 * Composition of mappings
 * Some further results and examples on mappings


 * Semigroups
 * Introduction
 * Binary operations
 * Associativity and commutativity
 * Semigroups: definition and examples
 * Powers of an element in a semigroup
 * Identity elements and inverses


 * An Introduction to Groups
 * The definition of a group
 * Examples of groups
 * Elementary consequences of the group axioms
 * Subgroups
 * Some important general examples of subgroups
 * Period of an element
 * Cyclic groups


 * Cosets and Lagrange's theorem
 * Introduction
 * Multiplication of the subsets of a group
 * Another approach to cosets
 * Lagrange's theorem
 * Some consequences of Lagrange's theorem


 * Homomorphisms, normal subgroups and quotient groups
 * Introduction
 * Isomorphic groups
 * Homomorphisms and their elementary properties
 * Conjugacy
 * Normal subgroups
 * Quotient groups
 * The quotient group $$G / Z$$
 * The first isomorphism theorem


 * Rings
 * Introduction
 * The definition of a ring and its elementary consequences
 * Special types of ring and ring elements
 * Subrings and subfields
 * Ring homomorphisms
 * Ideals
 * Principal ideals in a commutative ring with a one
 * Factor rings
 * Characteristic of an integral domain or field
 * Factorization in an integral domain
 * Construction of fields as factor rings
 * Polynomial rings over an integral domain
 * Some properties of $$F \left[{X}\right]$$, where $$F$$ is a field