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Fundamental Concepts of Abstract Algebra PDF

529 Pages·2011·8.22 MB·English
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Fundamental Concepts of Abstract Algebra G E ERTURDE HRLICH The University of Maryland D P , I . OVER UBLICATIONS NC Mineola, New York Copyright Copyright © 1991 by Gertrude Ehrlich. All rights reserved. Bibliographical Note This Dover edition, first published in 2011, is an unabridged republication of the work originally published in 1991 by PWS-Kent Publishing Company, Boston. Library of Congress Cataloging-in-Publication Data Ehrlich, Gertrude. Fundamental concepts of abstract algebra / Gertrude Ehrlich. — Dover ed. p. cm. Originally published: Boston : PWS-Kent Pub., 1991. Includes bibliographical references and index. ISBN-13: 978-0-486-48589-8 (pbk.) ISBN-10: 0-486-48589-7 (pbk.) 1. Algebra, Abstract. I. Title. QA162.E37 2011 512'.02—dc23 2011019127 Manufactured in the United States by Courier Corporation 48589701 www.doverpublications.com ″All right,″ said the Cat; and this time it vanished quite slowly, beginning with the end of the tail, and ending with the grin, which remained some time after the rest of it had gone. ″Well! I′ve often seen a cat without a grin,″ thought Alice, ″but a grin without a cat! It’s the most curious thing I ever saw in all my life!″ (Lewis Carroll, Alice in Wonderland) Preface his book is intended as a text in abstract algebra for undergraduate mathematics majors. It contains ample material for a two-semester sequence, including a thorough treatment of field theory during the second semester. Some knowledge of the rudiments of linear algebra is assumed from the start, and a fair number of examples are drawn from this area. In Sections 10–17 of Chapter 3, finite-dimensional vector spaces are treated in an abstract algebraic setting. These sections may be used for review, for reference, or to enhance the linear algebra background of the students. Some possible two- semester sequences are described below. The exercise list in each section of each chapter starts with an exercise consisting of True-or-False statements, selected with the aim of inducing a thoughtful review of materials covered in the section, and dispelling misconceptions at an early stage. Possible Two-Semester Sequences: Model 1 (including a full treatment of Galois Theory) First Semester: Chapter 1 (1–3) Chapter 2 (1–11; 12, 13 if time permits) Chapter 3 (1–9; 10 if time permits) Second Semester: Chapter 2 (14, 15) Chapter 4 (1–17) Model 2 (including some field theory, but not Galois theory, and an introduction to abstract linear algebra) First Semester: Chapter 1 (1–3) Chapter 2 (1–11) Chapter 3 (1–8) Second Semester: Chapter 2 (12–15) Chapter 3 (9–17) Chapter 4 (1–5) Acknowledgements am deeply indebted to my colleagues, Professors John Horvath, Adam Kleppner, and David Schneider, who taught from the manuscript and made many valuable comments; to numerous students, notably Greg Grant and Aaron Naiman (champion error finders); and to Kristi Aho Kraft and Virginia Sauber Vargas who typed a large portion of the manuscript, with much skill and forbearance. I would also like to express my appreciation to the following reviewers: Daniel D. Anderson (University of Iowa), Ida Z. Arms (Indiana University of Pennsylvania), Robert Beezer (University of Puget Sound), Patrick J. Costello (Eastern Kentucky University), John C. Higgins (Brigham Young University), Frederick Hoffman (Florida Atlantic University), William J. Keane (Boston College), Daniel B. Shapiro (Ohio State University), Anita A. Solow (Grinnell College), John R. Stallings (University of California-Berkeley), Paul M. Weichsel (University of Illinois), and Anne Ludington Young (Loyola College). Glossary of Symbols The set of all integers, rational numbers, real numbers, and complex numbers, respectively. The set of all positive integers, rational numbers, and real numbers, respectively. The set of all non-zero rational numbers, real numbers, and complex numbers, respectively. M (A) The set of all n × n matrices with entries in A. n GL ( ) The set of all non-singular n × n matrices with real entries. n SL ( ) The set of all n × n matrices with real entries and with n determinant equal to 1. Is a subset of. Contains as a subset. Is an element of. ∈ f : A → B f is a function from A to B. Contents Preliminaries 1.1 Introduction 1.2 Sets, Relations, and Functions 1.3 The Integers Groups 2.1 Binary Operations 2.2 Groups 2.3 Subgroups, Cyclic Groups, and the Order of an Element 2.4 Isomorphism 2.5 Cosets and Lagrange’s Theorem 2.6 Permutation Groups 2.7 Cayley’s Theorem; Geometric Groups 2.8 Normal Subgroups 2.9 Homomorphism, Factor Groups, and the Fundamental Theorem of Homomorphism for Groups 2.10 Further Isomorphism Theorems; Simple Groups 2.11 Automorphisms and Invariant Subgroups 2.12 Direct Products of Groups 2.13 The Structure of Finite Abelian Groups 2.14 Solvable Groups 2.15 Primary Groups and the Sylow Theorems

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A text in abstract algebra for undergraduate mathematics majors, this volume contains enough material for a two-semester course. It presents extensive coverage of set theory, groups, rings, modules, vector spaces, and fields. Examples, definitions, theorems, and proofs appear throughout, along with
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