A Book of Abstract Algebra - UMD

A BOOK OF ABSTRACT ALGEBRA

Second Edition Charles C. Pinter

Professor of Mathematics Bucknell University

Dover Publications, Inc., Mineola, New York

Copyright ? 1982, 1990 by Charles C. Pinter All rights reserved.

Copyright

Bibliographical Note

This Dover edition, first published in 2010, is an unabridged republication of the 1990 second edition of the work originally published in 1982 by the McGraw-Hill Publishing Company, Inc., New York.

Library of Congress Cataloging-in-Publication Data

Pinter, Charles C, 1932? A book of abstract algebra / Charles C. Pinter. -- Dover ed. p. cm. Originally published: 2nd ed. New York : McGraw-Hill, 1990. Includes bibliographical references and index. ISBN-13: 978-0-486-47417-5 ISBN-10: 0-486-47417-8 1. Algebra, Abstract. I. Title.

QA162.P56 2010 512.02--dc22

Manufactured in the United States by Courier Corporation 47417803



2009026228

To my wife, Donna, and my sons,

Nicholas, Marco, Andr?s, and Adrian

CONTENTS*

Preface

Chapter 1 Why Abstract Algebra? History of Algebra. New Algebras. Algebraic Structures. Axioms and Axiomatic Algebra. Abstraction in Algebra.

Chapter 2 Operations Operations on a Set. Properties of Operations.

Chapter 3 The Definition of Groups Groups. Examples of Infinite and Finite Groups. Examples of Abelian and Nonabelian Groups. Group Tables. Theory of Coding: Maximum-Likelihood Decoding.

Chapter 4 Elementary Properties of Groups Uniqueness of Identity and Inverses. Properties of Inverses. Direct Product of Groups.

Chapter 5 Subgroups Definition of Subgroup. Generators and Defining Relations. Cay ley Diagrams. Center of a Group. Group Codes; Hamming Code.

Chapter 6 Functions Injective, Surjective, Bijective Function. Composite and Inverse of Functions. Finite-State Machines. Automata and Their Semigroups.

Chapter 7 Groups of Permutations Symmetric Groups. Dihedral Groups. An Application of Groups to Anthropology.

Chapter 8 Permutations of a Finite Set Decomposition of Permutations into Cycles. Transpositions. Even and Odd Permutations. Alternating Groups.

Chapter 9 Isomorphism The Concept of Isomorphism in Mathematics. Isomorphic and Nonisomorphic Groups. Cayley's Theorem.

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