Undergraduate Texts in Mathematics

Editors

F. W. Gehring P. R. Halmos

Advisory Board

C. DePrima I. Herstein

Undergraduate Texts in Mathematics

Apostol: Introduction to Analytic Number Theory. 1976. xii, 338 pages. 24 ilIus.

Armstrong: Basic Topology. 1983. xii, 260 pages. 132 ilIus.

Bak/Newman: Complex Analysis. 1982. x, 224 pages. 69 ilIus.

Banchoff/Wermer: Linear Algebra Through Geometry. 1983. x, 257 pages. 81 illus.

Childs: A Concrete Introduction to Higher Algebra. 1979. xiv, 338 pages. 8 iIIus.

Chung: Elementary Probability Theory with Stochastic Processes. 1975. xvi, 325 pages. 36 ilIus.

Croom: Basic Concepts of Algebraic Topology. 1978. x, 177 pages. 46 iIIus.

Curtis: Linear Algebra: An Introductory Approach. 1984. x, 337 pages. 37 illus.

Dixmier: General Topology. 1984. x, 140 pages. 13 iIIus.

Driver: Why Math? 1984. xiv, 234 pages. 87 iIIus.

Ebbinghaus/Flum/Thomas Mathematical Logic. 1984. xii, 216 pages. I iIIus.

Fischer: Intermediate Real Analysis. 1983. xiv, 770 pages. 100 iIIus.

Fleming: Functions of Several Variables. Second edition. 1977. xi, 411 pages. 96 ilIus.

Foulds: Optimization Techniques: An Introduction. 1981. xii, 502 pages. 72 ilIus.

Foulds: Combinatorial Optimization for Undergraduates. 1984. xii, 222 pages. 56 ilIus.

Franklin: Methods of Mathematical Economics. Linear and Nonlinear Programming. Fixed-Point Theorems. 1980. x, 297 pages. 38 iIIus.

Halmos: Finite-Dimensional Vector Spaces. Second edition. 1974. viii, 200 pages.

Halmos: Naive Set Theory. 1974, vii, 104 pages.

looss/ Joseph: Elementary Stability and Bifurcation Theory. 1980. xv, 286 pages. 47 iIIus.

Janich: Topology. 1984. ix, 180 pages (approx.). 180 iIIus.

Kemeny/Snell: Finite Markov Chains. 1976. ix, 224 pages. II iIIus.

Lang: Undergraduate Analysis. 1983. xiii, 545 pages. 52 iIIus.

Lax/Burstein/Lax: Calculus with Applications and Computing, Volume I. Corrected Second Printing. 1984. xi, 513 pages. 170 iIIus.

LeCuyer: College Mathematics with A Programming Language. 1978. xii, 420 pages. 144 illus.

continued after Index

Rudolf Lidl Gunter Pilz

Applied Abstract Algebra

With 175 Illustrations

Springer-Verlag New York Berlin Heidelberg Tokyo

Rudolf Lidl Department of Mathematics University of Tasmania Hobart, Tasmania, 7001 Australia

Editorial Board

F. W. Gehring Department of Mathematics University of Michigan Ann Arbor, MI 48109 U.S.A.

Giinter Pilz Institut fiir Mathematik Universitat Linz A-4040 Linz Austria

P. R. Halmos Department of Mathematics Indiana University Bloomington, IN 47405 U.S.A.

AMS Subject Classifications: 05-01, 06-01, 08-01, 13-01, 15-01, 94A24, 05BXX, 20B25, 68BIO, l2C99, 94CXX, 68030, 68FIO, 20M35

Library of Congress Cataloging in Publication Data Lidl, Rudolf.

Applied abstract algebra. (Undergraduate texts in mathematics) Bibliography: p. Includes indexes. 1. Algebra, Abstract. I. Pilz, Giinter.

II. Title. III. Series. QA162.L53 1984 512'.02 84-10576

© 1984 by Springer-Verlag New York Inc. Softcover reprint of the hardcover 1st edition 1984

All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer-Verlag, 175 Fifth Avenue, New York, New York 10010, U.S.A.

Typeset by J. W. Arrowsmith Ltd., Bristol, England.

9 8 7 6 5 432 I

ISBN-13: 978-0-387-96166-8 DOl: 10.1007/978-1-4615-6465-2

e-ISBN-13: 978-1-4615-6465-2

To Pamela and Gerti

Preface

There is at present a growing body of opinion that in the decades ahead discrete mathematics (that is, "noncontinuous mathematics"), and therefore parts of applicable modern algebra, will be of increasing importance. Cer­tainly, one reason for this opinion is the rapid development of computer science, and the use of discrete mathematics as one of its major tools.

The purpose of this book is to convey to graduate students or to final-year undergraduate students the fact that the abstract algebra encountered pre­viously in a first algebra course can be used in many areas of applied mathematics. It is often the case that students who have studied mathematics go into postgraduate work without any knowledge of the applicability of the structures they have studied in an algebra course.

In recent years there have emerged courses and texts on discrete mathe­matics and applied algebra. The present text is meant to add to what is available, by focusing on three subject areas. The contents of this book can be described as dealing with the following major themes:

Applications of Boolean algebras (Chapters 1 and 2). Applications of finite fields (Chapters 3 to 5). Applications of semigroups (Chapters 6 and 7).

Each of these three themes can be studied independently. We have not tried to write a comprehensive book on applied algebra, rather we have tried to highlight some algebraic structures which seem to have most useful applica­tions. Each of these topics is relevant to and has strong connections with computer science.

We assume that the reader has the mathematical maturity of a beginning graduate student or of a last-year undergraduate student at a North American university or of a third or final-year Bachelor or Honors student

viii Preface

in the United Kingdom or Australia. Thus the text is addressed mainly to a mature mathematics student and should also be useful to a computer scientist or a computer science student with a good background in algebra. Some students and lecturers might also be interested in seeing some not-so­well-known applications of selected algebraic structures. The reader is expected to be familiar with basic ideas about groups, rings, fields and linear algebra as the prerequisites for this book. All these requirements 'are met in a first course on linear algebra and an introductory course on abstract algebra.

The first topic, treated in Chapters I and 2, deals with properties and applications of Boolean algebras and their use in switching circuits and simplification methods. The next three chapters, which form the core of the text, comprise properties and applications of finite fields. Considerable emphasis is given to computational aspects. Chapter 3 contains the basic properties of finite fields and polynomials oVer finite fields; these will be used in the following two chapters. Chapter 4 contains topics from algebraic coding theory with a decoding procedure for BCH codes as its climax. Chapter 5 is devoted to other areas of applications of rings and finite fields, such as combinatorics, cryptography and linear recurring sequences. The third major topic, applications of semigroups to automata, formal languages, biology and sociology, is covered in Chapters 6 and 7.

Throughout the text, great emphasis is put on computational examples in the belief that most readers learn to do mathematics by solving numerical problems. A number of problems is given at the end of each section. Each paragraph ends with a number of exercises which are solved in Chapter 8 of this book. It is hoped that the reader will work through these problems and exercises and use the solutions in Chapter 8 only as a check of their understanding of the material.

The appendix consists of two parts. Part A contains fundamental defini­tions and properties of sets, logical symbols, relations, functions and alge­braic operations. More on that can be found in almost every introductory text. Part B contains some computer programs to perform some of the algorithms presented in the text. The advent of microcomputers and the wide and rapidly increasing availability of desk-top computers prompted us to do so. In certain areas of applied mathematics, the computer is an indispensible tool.

The chapters are divided into sections; larger sections are subdivided into subsections A, B, etc. References in the text are organized such that 1.3.5 refers to item (Theorem, Definition, ... ) number 5 in section 3 of Chapter l. Within one chapter we use the abbreviation 3.5 to refer to item number 5 in section 3 of the present chapter. We refer to items in the Bibliography by writing the author's name in small capitals. The symbol D denotes the end of a proof or an example. Some of the more difficult or not quite straightforward problems, exercises or whole sections are marked with an asterisk *. The notes at the end of each chapter provide some

Preface IX

historical comments and references for further reading. Parts of the material of this book appeared (along with some other applications of algebra) in the authors' German Text Angewandte Abstrakte Algebra, Vols. I, II (Bib­liographisches Institut, Mannheim, 1982).

It is with pleasure that we thank friends and colleagues for helpful suggestions after critically reviewing parts of the manuscript. We gratefully acknowledge contributions to the final draft by: Elizabeth J. Billington (Brisbane, Australia); Donald W. Blackett (Boston, Massachusetts); Henry E. Heatherly (Lafayette, Louisiana); Carlton J. Maxson (College Station, Texas); John D. P. Meldrum (Edinburgh, Scotland); Ken Miles (Mel­bourne, Australia); Alan Oswald (Teesside, England) and Peter G. Trotter (Hobart, Australia). Finally, we wish to thank the editorial and production staff of Springer-Verlag for their kind cooperation throughout the prep­aration of this book.

March 1984 R. L. and G. P.

Contents

List of Symbols

CHAPTER 1 Lattices § I. Properties of Lattices

A. Lattice Definitions B. Modular and Distributive Lattices

§2. Boolean Algebras A. Basic Properties B. Boolean Polynomials, Ideals

§3. Minimal Forms of Boolean Polynomials Notes

CHAPTER 2

Applications of Lattices § l. Switching Circuits

A. Basic Definitions B. Applications of Switching Circuits

§2. Propositional Logic *§3. Further Applications Notes

CHAPTER 3

Finite Fields and Polynomials § l. Rings and Fields

A. Rings, Ideals, Homomorphisms B. Polynomials C. Fields

*D. Algebraic Extensions

xv

14 23 23 31 45 53

56 56 56 67

100 107 118

120 120 120 127 132 136

xu

§2. Finite Fields §3. Irreducible Polynomials over Finite Fields §4. Factorization of Polynomials over Finite Fields §5. The Nullspace of a Matrix (Appendix to §4)

Notes

CHAPTER 4

Coding Theory § I. Linear Codes §2. Cyclic Codes

*§3. Special Cyclic Codes Notes

CHAPTER 5

Further Applications of Fields and Groups § I. Combinatorial Applications

A. Hadamard Matrices B. Balanced Incomplete Block Designs C. Steiner Systems, Difference Sets and Latin Squares

§2. Algebraic Cryptography A. Single Key (Symmetric) Cryptosystems B. Public-Key Cryptosystems

§3. Linear Recurring Sequences §4. Fast Adding

*§5. P6lya's Theory of Enumeration Notes

CHAPTER 6

Contents

144 156 168 181 187

192 192 211 227 243

246 246 246 253 260 270 271 282 296 311 313 326

Automata 331 § I. Semi automata and Automata 331 §2. Description of Automata; Examples 333 §3. Semigroups 338

A. Fundamental Concepts 338 B. Subsemigroups, Homomorphisms 341 C. Free Semigroups 348

§4. Input Sequences 354 §5. The Monoid of a (Semi-) Automaton and the (Semi-) Automaton of a

Monoid 358 §6. Composition and Decomposition 361

A. Elementary Constructions of Automata 361 * B. Cascades 366

§7. Minimal Automata 371 Notes 376

CHAPTER 7

Further Applications of Semigroups 379 § l. Formal Languages 379

A. Approach via Grammar 380

Contents

B. Approach via Automata and Semigroups C. Connections Between the Different Approaches

§2. Semigroups in Biology §3. Semigroups in Sociology

A. Kinship Systems B. Social Networks

Notes

CHAPTER 8 Solutions to the Exercises

Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Chapter 7

Appendix A. Some Fundamental Concepts B. Computer Programs

Bibliography

Author Index

Subject Index

Xlll

385 387 390 397 397 400 407

409 409 425 446 462 480 494 501

505 505 509

522

535

539

List of Symbols

Symbols which are frequently used are listed in order of appearance in the text.

x

< V ¢:>

IR N sup inf n u ILl ~

[x, y] C M\A x'

cartesian product 2, 506 colon indicating definition 2, 506 empty set 2 the power set 2, 506 set-theoretic inclusion 2, 506 less than or equal 2 strictly less than 2 for all 4, 506 if and only if 4, 506 the set of real numbers 4, 505 the set of natural numbers 4, 505 supremum 4 infimum 4 meet 5 join 5 cardinality of L 6 implication 6, 506 interval II the set of complex numbers 14, 505 set-theoretic difference 19, 506 complement 20 the Boolean algebra of 0 and 1 23 Boolean isomorphic 27 embedding monomorphism 30

xvi

ker

the set of Boolean polynomials 32 Boolean polynomial function 32 the set of Boolean polynomial functions 33 equivalence of Boolean polynomials 33 the set of mappings from Minto B 34, 507 being ideal in 41 kernel 41

List of Symbols

1\ logical "and", conjunction 41, 101, 505 (b) principal ideal 41 3 there exists 46, 506 SQ standard square 64 HA half-adder 82 -, v

{}

char R IP III ==n or == [a] Imh R/I Hom(R, R') PIO R[[x]] R[x] gcd UFO peR) F(a) [K: F] I' fp" Qn IL ind Iq(k)

ordl I(q, n; x)

r

EB i=1

negation 10 1, 506 disjunction, logical "or" 101, 505 subjunction 102 bijunction 102 sample space III characteristic of R 121 the set of primes 122, 505 the set of rational numbers 122, 505 congruence relation mod n on Z 123 residue class of a mod n 123 image of h 124 factor ring 124 the set of all homomorphisms from R into R' 125 principal ideal domain 125 the ring of formal power series over R 127 the polynomial ring over R 127 greatest common divisor 128 unique factorization domain 129 the set of polynomial functions over R 129 the simple extension of a field F 135 the degree of the field K over the field F 137 the formal derivative of I 140 the finite field with p" elements 146 the nth cyclotomic polynomial 149 the Mobius IL-function 150 index or discrete logarithm 153 the number of all monic, irreducible polynomials of degree k over f q 159 the order of a polynomial 160 the product of all monic irreducible polynomials in f q[x] of degree n 166

direct sum 169

List of Symbols

h 0 x q

Cs (A, I) d(x,y) w(x) dmin

Sr(x) C-L

S(y) Cm

X Vn Z(v) lcm DFT BCH RS QR RM(m, d) I n

S(t, k, v) PG(2, q) J(a, n)

r xl Sn Z(Sn) sign( 7T)

An K(G) Orb(x) Stab(x) Fix(g) 8 A a 91l(M) Gs

R' S S d(S" S2) si,:s siz si, X si2

si,*siz si, x~ si2

substitution of polynomials 170 cyclotomic coset 177

XVll

matrix consisting of a matrix A and the identity matrix I 194 Hamming distance 197 Hamming weight 197 minimum distance 197 the sphere of radius r about x 198 dual code 199 syndrome 201,219 the Hamming code of length m 203 character 207 the vector space of polynomials over IF q of degree < n 212 cyclic shift 212 least common multiple 217 discrete Fourier transform 222 Bose, Chaudhuri, Hocquenghem 229 Reed, Solomon 231 quadratic residue 238 Reed-Muller code 241 n x n matrix of ones 251 Steiner system 260 the projective geometry 260 the Jacobi symbol 296 smallest natural number 2:: x 312 the symmetric group on {I, 2, ... , n} 314 the center of Sn 315 signature of 7T 315 alternating groups 316 commutator group 316 orbit of x 317 stabilizer of x 317 319 next-state function 331 output function 332 symmetric difference 338, 506 set of relations on M 339 group kernel 342 transitive hull 343 least common multiple of semigroups 352 greatest common divisor of semigroups 352 distance between S, and S2 353 si, is sub automaton of siz 361 direct product of automata 364 series composition of automata 365 cascade of automata 366

xviii

SIT S~T

-k

A* A* C§

(X,Y)E~

~ ~t

L(C§) l(x) 8(0 .. O2 )

1(1 )n id, t

XAi Fn

Fn A~ GL(n, A)

the semigroup S divides T 368 the wreath product 368 k-equivalence 372 the free semi group over a set A 380 the free monoid over a set A 380 grammar 380 productions 380 extension of ~ 380 380 the language generated by C§ 380 length of x 383 distance between societies 399 means 1:5 i :5 n 445 identity function 507 the cartesian product of Ai 508

List of Symbols

the vector space of row vectors (x .. ... , Xn), Xi E F 509 the vector space of column vectors (Xi> •.• , Xn) T, Xi E F 509 the set of n x n matrices with entries in A 509 the general linear group 509