Topics in Applied Abstract Algebra

335 Pages·2004·5.28 MB·English

by  S. R. Nagpaul

Topics in
Applied
Abstract
Algebra
S. R. Nagpaul
Jain
S. K.
THE BROOKS/COLE SERIES IN
ADVANCED MATHEMATICS
Paul J. Sally, Jr. . mrroR
TOPICS IN
APPLIED
ABSTRACT
ALGEBRA
The Brooks/Cole Series in Advanced Mathematics
Paul J. Sally, Jr., Editor
Series editor Paul J. Sally, Jr. and Brooks/Cole have developed this prestigious list
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TOPICS IN
APPLIED
ABSTRACT
ALGEBRA
S.R. Nagpaul
Ohio University
S.K. Jain
Ohio University
THC>I•VIS C>N
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CONTENTS
Preface ix
0
PRELIMINARY ALGEBRAIC CONCEPTS 1
0.1 Sets, Mappings, Relations, and Binary Operations
0.1 .1 Sets 1
0.1 .2 Mappings 4
0.1 .3 Relations 5
0.1.4 Binary Operations 9
Exercises 0.1 9
0.2 Groups and Semigroups 11
0.2.1 Groups of Integers Modulo n 12
0.2.2 Groups of Matrices and Permutations 16
Exercises 0.2 1 7
0.3 Cyclic Groups, Order of an Element, and Direct Product 17
0.3.1 Order of an Element 17
0.3.2 Cyclic Groups 20
0.3.3 Direct Product of Two Groups 22
Exercises 0.3 23
0.4 Subgroups of a Group 23
Exercises 0.4 26
0.5 Quotient Groups and Homomorphisms 27
0.5.1 Quotient Groups 27
M.2 Homomorphisms of Groups 28
Exercises 0.5 31
0.6 Applications of Groups in Number Theory 31
0.7 Rings and Fields 34
0.7.1 Subrings, Ideals, and Quotient Rings 36
Exercises 0.7 38
0.8 Finite Fields 39
Exercises 0.8 46
v
Vi CONTENTS
1
BOOLEAN ALGEBRAS AND SWITCHING CIRCUITS 47
1 .1 Boolean Algebras 47
Exercises 1 .1 49
1 .2 Switches and Logic Gates 49
1 .2 .1 Switches 49
1 .2 .2 Logic Gates 52
1 .3 Laws of Boolean Algebra 53
1.3.1 New Notation 56
Exercises 1.3 57
1.4 Boolean Polynomials and Boolean Functions 57
1.4.1 Boolean Polynomials 57
1.4.2 Canonical Forms of a Boolean Polynomial 59
1 .4.3 Boolean Functions 61
Exercises 1 .4 64
1.5 Switching Circuits and Gate Networks 64
1.5.1 Switching Circuits 64
1.5.2 Gate Networks 67
1.6 Simplification of Circuits 69
Exercises 1.6 71
1.7 Designing Circuits 73
1.7.1 Switching Function of Circuits 73
1.7.2 Exclusive-OR Gate, Half-Adder, and Adder 77
1.8 Bridge Circuits 82
Exercises 1.8 85
2
BALANCED INCOMPLETE BLOCK DESIGNS 87
2.1 Basic Definitions and Results 87
2.2 Incidence Matrix of a BIBO 92
Exercises 2.2 96
2.3 Construction of BIBDs from Difference Sets 98
2.4 Construction of BIB Os Using Quadratic Residues 100
2.5 Difference Set Families 105
2.6 Construction of BIB Os from Finite Fields 106
Exercises 2 .6 111
2.7 Construction of BIB Os from Nearrings 112
2.8 Planar Nearrings 116
2.9 Finite Integral Planar Nearrings and BIBDs 120
2.10 Finite Fields and Planar Nearrings 124
Exercises2.10 127
Vii
CONTENTS
3
ALGEBRAIC CRYPTOGRAPHY 128
3.1 Substitution Ciphers 128
Exercises 3.1 134
3.2 Algebraic Enciphering Algorithms and Classical Cryptosystems 135
3.2.1 Modular Enciphering and Affine Cipher 135
3.2.2 Hill Cipher 142
Exercises 3 .2 144
3.3 Block Ciphers and Advanced Encryption Standard 145
3.3.1 Rijndael Algorithm 149
3.4 Public-Key Cryptosystems 152
3.4.1 Knapsack Cryptosystem 154
3.4.2 RSA Cryptosystem 157
Exercises 3.4 159
4
CODING THEORY 160
4.1 Introduction to Error-Correcting Codes 160
Exercises 4.1 166
4.2 Linear Codes 167
4.2.1 Generator and Parity-Check Matrices 168
4.2.2 Minimum Distance 174
4.2.3 Hamming Codes 176
4.2.4 Decoding 178
Exercises 4.2 182
4.3 Cyclic Codes 183
Exercises 4.3 196
4.4 BCH Codes 196
Exercises 4.4 207
5
SYMMETRY GROUPS AND COLOR PATTERNS 208
5.1 Permutation Groups 208
5.1.1 Even and Odd Permutations 213
Exercises 5.1 214
5.2 Groups of Symmetries 214
Exercises 5.2 219
5.3 Colorings and Color Patterns 220
5.4 Action of a Group on a Set 221
5.5 Burnside Theorem and Color Patterns 225
Exercises 5.5 234
Viii
CONTENTS
5.6 Polya's Theorem and Pattern Inventory 235
5.6.1 Polya's Theorem 236
5.6.2 Pattern Inventory 239
5.6.3 Cycle Index Polynomial 240
Exercises 5.6 242
5.7 Generating Functions for Nonisomorphic Graphs 243
Exercises 5.7 248
6
WALLPAPER PATTERN GROUPS 249
6.1 Group of Symmetries of a Plane 249
6.2 Wallpaper Pattern Groups 253
6.3 Change of Basis in JR2 257
6.4 Point Groups and Lattice Types 260
6.5 Equivalence of WP Groups 264
6.6 Classification of Point Groups 267
6.7 Classification of WP Groups 274
6.8 Sample Patterns 290
Exercises 6.8 290
Answers and Hints for Selected Exercises 295
Index 318
PREFACE
This book presents some interesting applications of abstract algebra to practical
real-world problems. Whereas many applications of calculus are presented in un
dergraduate courses, usually no such applications are given in courses on abstract
algebra. The object of this book is to fill this lacuna. It is hoped that this will make
the study of abstract algebra more interesting and meaningful, especially for those
whose interest in algebra is not confined to mere abstract theory. Among the ap
plications discussed in the book are designing and simplifying switching circuits
and gate networks used in computer science; designing block designs to conduct
statistical experiments for unbiased study of samples; designing secret-key cryp
tosystems and public-key cryptosystems for secure transmission of sensitive or
secret data; designing error-correcting codes for transmission of data through noisy
channels; computing the number of color patterns of a given design; and classifying
the symmetry groups of wallpaper patterns.
The book may be used as a text for senior-level and beginning graduate-level
students for a course in applied abstract algebra. It is addressed to two categories of
students: (1) those who are majoring in mathematics and are interested in knowing
about applications of what they have learned in an abstract algebra course, and (2)
those who are majoring in other disciplines like physics, computer science, and
engineering, and deal with these applications in their area of study but often do
not have sufficient understanding of the mathematics involved. Of course, the book
may also be used profitably as a supplementary text for a regular course in abstract
algebra or as a reference book for all scientists in general. This book may indeed
be looked upon as a companion volume to all books in abstract algebra.
The book is self-contained and does not assume a prior knowledge of abstract
algebra, although a knowledge of elementary linear algebra is taken for granted.
The algebraic theory used in the applications is fully developed here. However,
this book is not meant to be used as a text for a course in abstract algebra per se.
The amount of algebraic theory presented here is just what is required for the
applications discussed in the book.
The opening chapter, called Chapter 0, gives a somewhat-condensed account
of the basic algebraic systems-namely, groups, rings, and fields, and their salient
properties. We have omitted the proofs of several theorems in this chapter that are
quite easy to prove. A student who has already done a course in abstract algebra
ix