Applications of Abstract Algebra

200 Pages·1985·7.175 MB·English

by  George Mackiw

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Anplications
of Abstract
Algebra
George Mackiw
John Wiley & Sons
New York / Chichester / Brisbane / Toronto / Singapore
863596
Copyright (c)1985 by John Wiley & Sons, Inc.
All rights reserved.
Reproduction or translation of any part of
this work beyond that permitted by Section
107 or 108 of the 1976 United States Copyright
Act without the permission of the copyright
owner is unlawful. Requests for permission
or further information should be addressed to
the Permissions Department, John Wiley & Sons, Inc.
ISBN 0-471-81078-9
Printed in the United States of America
10 ORS sOrar aes 2s 1
Preface
Abstract or modern algebra is considered by many to be one of
the more beautiful, elegant and coherent branches of mathematics. Yet,
in spite of the greater emphasis today on the use of discrete methods,
the study of groups, rings and fields seems to have suffered. A primary
reason may be that traditional presentations of the subject have not
placed much emphasis on motivation and applications, providing perhaps
less reason for the average student to be attracted to or to study the
subject. For example, while it is often mentioned that symmetry groups
are important in chemistry and physics, rarely are concrete examples of
such groups given in introductory texts.
The goal of this text is to provide a collection of
applications of abstract algebra to a student who is currently taking a
One or two semester introduction to the basics of groups, rings and
fields. The text is thus intended for use as a supplement to an
Abstract Algebra course, though it certainly may also be used in a
separate course on applications or as a reference. The applications
included here cover exact computing, error-correcting codes, the
construction of block designs useful in statistics, crystallography,
integer programming, cryptography and combinatorics. A more detailed
description of individual chapters follows below.
My motivation in writing this text was to present a series of
applications that would be accessible to the student in a first course,
provide examples of algebra in use, contain topics not previously
gathered together in text form, and hopefully be interesting. Topics
such as classical Greek constructions and applications of Galois Theory
have deliberately been omitted since these are precisely the ones that
are covered in many excellent texts. Applications to quantum mechanics
and physics are also absent since they tend to require a background
deeper than that possessed by the beginning student.
The chapters that follow are, with two exceptions, independent
of each other. We include a description of the content and
prerequisites for each chapter.
1) Computing in Zu - Exact Solutions of Linear Equations
Problems of round-off and precision are inherent in the
floating-point arithmetic used in computers. This chapter describes how
the arithmetic of the ring Zy and the Chinese Remainder Theorem may be
helpful in avoiding such problems when dealing with systems of linear
equations.
Prerequisites: elementary linear algebra, modular arithmetic, ring
homomorphisms.
2) Block Designs
The use of abelian groups, finite fields and rings in the
construction of block designs, structures which are useful in the design
of experiments.
Prerequisites: elementary theory of abelian groups, modular arithmetic,
the fields GF(p"), primitive elements.
3) Error Correcting Codes I: Hamming Codes
An introduction to the basics of error correcting codes. This
chapter features a detailed development of the Hamming codes and the
rationale behind their error correcting capabilities.
Prerequisites: elementary linear algebra, the field GF(2), cosets, the
fundamental theorem of group homomorphisms.
4) Error Correcting Codes II: BCH Codes
This requires the previous chapter and features the
development of the multiple error correcting BCH codes. It includes a
step by step exposition of the error correcting procedures attached to
BCH codes, which serve as a wonderful primer for reinforcing the
arithmetic of finite fields.
Prerequisites: elementary linear algebra, the finite fields GF(2"),
primitive elements, minimal polynominals.
5) Crystallographic Groups in the Plane I
This serves as an introduction to the classification of
crystals via their symmetry groups. Introduces the notion of
crystallographic space groups, translation subgroups and point groups in
two dimensions.
Prerequisites: elementary linear algebra and group theory, quotient
groups, fundamental theorem of group homomorphisms.
6) Appendix: Orthogonal Matrices in Two Dimensions
Develops from scratch results on 2 x 2 orthogonal matrices
which are needed in the above chapter.
Prerequisites: familiarity with 2 x 2 matrices and determinants.
7) Crystallographic Groups in the Plane II
This is a continuation of the previous chapter and provides a
rigorous development of the fact that there are only 17 inequivalent
crystallographic groups in the plane. These are illustrated by concrete
matrix representations and drawings.
Prerequisites: “same as the previous chapter.
8) The RSA Public Key Cryptosystem
Discusses the issues behind modern public key cryptography and
develops the algebra underlying the Rivest-Shamir-Adleman system.
Includes a discussion of recent developments in primality testing and
factorization.
Prerequisites: modular arithmetic, the ring Zy, Lagrange's Theorem for
groups.
9) Integer Programming
Introduces integer programming through optimization
problems. Develops the role finite abelian groups play in the solution
of integer programming problems.
Prerequisites: elementary linear algebra and linear programming, use of
the simplex algorithm, group homomorphisms, quotient groups, the
exponent of a finite abelian group.
10) Group Theory and Counting
The role group theory plays in combinatorial problems;
includes a derivation of Burnside's Lemma with many concrete examples.
Prerequisites: elementary group theory, cosets, equivalence relations.
As may be evident, elementary linear algebra is invoked
throughout. This should present no significant difficulty, since it is
now commonplace for most students to encounter linear algebra in either
freshman or sophomore year. With the exception of the two chapters each
on crystallography and error correcting codes, the individual sections
of the text are independent of each other and may be used in any order
desired. Each chapter includes a set of exercises and a list of further
references.
I should like to thank the staff of the Loyola College
Communication Center, especially Charlette Hinchliffe, Nancy Marshall,
Mary Muhler, and Marion Weilgosz for their diligent work in preparing
this manuscript. Dean David Roswell was very helpful in his support and
encouragement. It was my pleasure to engage in many interesting
conversations with Professor Alan Goldman of Johns Hopkins University
during a sabbatical year spent at the Center for Applied Mathematics at
the National Bureau of Standards. It was he who provided the idea for
the chapter on integer programming. It is my hope that both students
and teachers of abstract algebra will find this collection useful.
George Mackiw
Baltimore, Maryland
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