Book:DummitFoote
This is a book clip page. When referencing the book, transclude this page using Template:BooklinkAbstract Algebra by David S. Dummit and Richard M. Foote, 10-digit ISBN 0471433349, 13-digit ISBN 978-0471433347
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This is the third edition. Note that page numbers may differ in earlier and later editions.
Contents
About this book
General information
Abstract Algebra is written by David S. Dummit and Richard M. Foote, both working at the University of Vermont. The book covers a wide range of basic algebraic topics. According to the authors (preface to the third edition), the basic theme of this book is: the power and beauty that accrues from a rich interplay between different areas of mathematics. The emphasis throughout has been to motivate the introduction and development of important algebraic concepts using as many examples as possible. [The authors] have tried to touch on many of the central themes in elementary algebra in a manner suggesting the very natural development of these ideas.
Relevant contents in group theory
Part 1 (Group Theory) contains most of the book's group theory content:
- Introduction to Groups
- Basic axioms and examples (pages 16-21; exercises on pages 21-23), includes the following:
- General definition of binary operation
- Definition of commutative binary operation
- Definition of associative binary operation
- Definition of group, with examples
- Proof that identity element and inverses in a group are unique (related material in this wiki is at: equivalence of definitions of group)
- Proof that Inverse map is involutive
- Proof that associative implies generalized associative
- Proof that in a group, equations have unique solutions (related material in this wiki is at: group implies quasigroup)
- Proof that in a group, cancellation laws hold (related material in this wiki is at: invertible implies cancellative in monoid)
- Dihedral groups
- Symmetric groups
- Matrix groups
- The quaternion group
- Homomorphisms and isomorphisms
- Group actions
- Basic axioms and examples (pages 16-21; exercises on pages 21-23), includes the following:
- Subgroups
- Definition and examples
- Centralizers and normalizers, stabilizers and kernels
- Cyclic groups and cyclic subgroups
- Subgroups generated by subsets of a group
- The lattice of subgroups of a group
- Quotient groups and homomorphisms
- Definitions and examples
- More on cosets and Lagrange's theorem
- The isomorphism theorems
- Composition series and the Holder program
- Transpositions and the alternating group
- Group actions
- Group actions and permutation representations
- Groups acting on themselves by left multiplication -- Cayley's theorem
- Groups acting on themselves by conjugation -- the class equation
- Automorphisms
- The Sylow theorems
- Simplicity of
- Direct and semidirect products in Abelian groups
- Direct products
- The fundamental theorem of finitely generated Abelian groups
- Table of groups of small order
- Recognizing direct products
- Semidirect products
- Further topics in group theory
- -groups, nilpotent groups and solvable groups
- Applications in groups of medium order
- A word on free groups
Style of definition
Most definitions that are introduced in-text are introduced formally, preceded by the word Definition. Typically, the block of definition text is followed by a list of examples, often from diverse fields. Some definitions are also introduced in the exercises, in a more informal manner. (This is in contrast with Artin, where a principal example often precedes the definition)
Style of proof
Most propositions stated formally in the main text are proved formally. A number of propositions are stated in exercises in the text at some point, and stated as theorems a little later in the text, with proof.