INTRODUCTION TO MODERN ALGEBRA PURE AND APPLIED MATHEMATICS A Program ofMonographs, Textbooks, and Lecture Notes EXECUTIVE EDITORS—MONOGRAPHS, TEXTBOOKs, AND LECTURE NOTES Earl J. Taft Rutgers University New Brunswick, New Jersey Edwin Hewitt University of Washington Seattle, Washington CHAIRMAN OF THE EDITORIAL BOARD S. Kobayashi University ofCalifornia, Berkeley Berkeley, California EDITORIAL BOARD Masanao Aoki W. S. Massey University ofCalifornia, Los Angeles Yale University Glen E. Bredon Irving Reiner Rutgers University University ofIllinois at Urbana-Champaign Sigurdur Helgason Paul J. Sally, Jr. Massachusetts Institute of Technology University ofChicago G. Leitman Jane Cronin Scan/on University ofCalifornia, Berkeley Rutgers University Marvin Marcus Martin Schechter University ofCalifornia, Santa Barbara Yeshiva University Julius L. Shaneson Rutgers University MONOGRAPHS AND TEXTBOOKS IN PURE AND APPLIED MATHEMATICS _i K. Yano, Integral Formulas in Riemannian Geometry (1970) . S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings (1970) V. S. Vladimirov, Equations ofMathematical Physics (A. Jeffrey, editor; A. Littlewood, translator) (1970) B. N. Pshenichnyi, Necessary Conditions for an Extremum (L. Neustadt, translation editor; K. Makowski, translator) (1971) L. Narici, E. Beckenstein, and G. Bachman, Functional Analysis and Valuation Theory (1971) D. S. Passman, Infinite Group Rings (1971) L. Dornhoff, Group Representation Theory (in two parts). Part A: Ordinary Representation Theory. Part B: Modular Representation Theory (1971, 1972) W. Boothby and G. L. Weiss (eds.), Symmetric Spaces: Short Courses Presented at Washington University (1972) Y. Matsushima, Differentiable Manifolds (E. T. Kobayashi, translator) (1972) 10. L. E. Ward, Jr., Topology: An Outline for a First Course (1972) out ofprl'nt 11. A. Babakhanian, Cohomological Methods in Group Theory (1972) 12. R. Gilmer, Multiplicative Ideal Theory (1972) 13. J. Yeh, Stochastic Processes and the Wiener Integral (1973) outofprint 14. J. Barros-Neto, Introduction to the Theory ofDistributions (1973) cutofprint 15. R. Larsen, Functional Analysis: An Introduction (1973) 16. K. Yano and S. Ishihara, Tangent and Cotangent Bundles: Differential Geometry (1973) 17. C. Procesi, Rings with Polynomial Identities (1973) 18. R. Hermann, Geometry, Physics, and Systems (1973) 19. N. R. Wallach, Harmonic Analysis on Homogeneous Spaces (1973) 20. J. Dieudonné, Introduction to the Theory of Formal Groups (1973) 21. I. Vaisman, Cohomology and Differential Forms (1973) 22. B.-Y. Chen, Geometry ofSubmanifolds (1973) 23. M Marcus, Finite Dimensional Multilinear Algebra (in two parts) (1973, 1975) 24. R. Larsen, Banach Algebras: An Introduction (1973) 25. R. 0. Kufala andA. L. Vitter (eds.), Value Distribution Theory: Part A; Part B. Deficit and Bezout Estimates by Wilhelm Stoll (1973) 26. K. B. Stolarsky, Algebraic Numbers and Diophantine Approximation (1974) 27. A. R. Magid, The Separable Galois Theory ofCommutative Rings (1974) 28. B. R. McDonald, Finite Rings with Identity (1974) 29. J. Satake, Linear Algebra (S. Koh, T. Akiba, and S. Ihara, translators) (1975) 30. J. S. Golan, Localization ofNoncommutative Rings (1975) 31. G. Klambauer, Mathematical Analysis (1975) 32. M K. Agoston, Algebraic Topology: A First Course (1976) 33. K. R. Goodearl, Ring Theory: Nonsingular Rings and Modules (1976) 34. L. E. Mansfield, Linear Algebra with Geometric Applications (1976) 35. N. J. Pullman, Matrix Theory and its Applications: Selected Topics (1976) 36. B. R. McDonald, Geometric Algebra Over Local Rings (1976) 37. C W. Groetsch, Generalized Inverses of Linear Operators: Representation and Approximation (1977) 38. J. E. KuczkowskiandJ. L. Gersting, Abstract Algebra: A First Look (1977) 39. C 0. Christenson and W. L. Voxman, Aspects ofTopology (1977) 40. M. Nagata, Field Theory (1977) 41. R. L. Long, Algebraic Number Theory (1977) 42. W. F. Pfeffer, Integrals and Measures (1977) 43. R. L. leeden and A. Zygmund, Measure and Integral: An Introduction to Real Analysis (1977) 44. J. H. Curtiss, Introduction to Functions ofa Complex Variable (1978) 45. K. Hrbacek and T. Jech, Introduction to Set Theory (1978) 46. W. S. Massey, Homology and Cohomology Theory (1978) 47. M Marcus, Introduction to Modern Algebra (1978) INTROgloJCTION MODERN ALGEBRA Marvin Marcus Institute for the Interdisciplinary ApplicationsofAlgebra and Combinatorics UniversityofCalifornia. Santa Barbara Santa Barbara, California MARCEL DEKKER, INC. New York and Basel Library ofCongress Catalogingin Publication Data Marcus, Marvin [Date] Introduction to modern algebra. (Monographs and textbooks in pure and applied mathematics ;v. 47) 1. Algebra, Abstract. I. Title. QA162.M37 512’.02 78-2482 ISBN 0-8247-6479-X COPYRIGHT © 1978 by MARCEL DEKKER, INC. ALL RIGHTS‘RESERVED Neither this book nor any part may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, microfilming, and recording, or by any information storage and retrieval system, without permission in writing from the publisher. MARCEL DEKKER, INC. 270 Madison Avenue, New York, New York 10016 Current printing (last digit): 10 9 8 7 6 5 4 3 2 l PRINTED IN THE UNITED STATES OF AMERICA To my wife Rebecca Elizabeth Marcus Contents Preface vii 1 . Basic Structures 1.1 Sets and Functions Exercises 14 Glossary 15 1.2 Algebraic Structures 16 Exercises 25 Glossary 30 1.3 Permutation Groups 31 Exercises 52 Glossary 54 2. Groups 56 2.] Isomorphism Theorems 56 Exercises 83 Glossary 86 2.2 Group Actions and the Sylow Theorems 86 Exercises 106 G10Ssary 109 2.3 Some Additional Topics 109 Exercises 123 Glossary 123 3. Rings and Fields 124 3.] Basic Facts 124 Exercises 136