Transformation groups for beginners

192 Pages·2004·1.175 MB·English

by  S. V. Duzhin, B. D. Chebotarevskii

Transformation Groups for Beginners
S. V. Duzhin
B. D. Tchebotarevsky

Contents
Preface 5
Introduction 6
Chapter 1. Algebra of points 11
1. Checkered plane 11
x
2. Point addition 13
x
3. Multiplying points by numbers 17
x
4. Centre of gravity 19
x
5. Coordinates 21
x
6. Point multiplication 24
x
7. Complex numbers 28
x
Chapter 2. Plane Movements 37
1. Parallel translations 37
x
2. Re(cid:13)ections 39
x
3. Rotations 41
x
4. Functions of a complex variable 44
x
5. Composition of movements 47
x
6. Glide re(cid:13)ections 52
x
7. Classi(cid:12)cation of movements 53
x
8. Orientation 56
x
9. Calculus of involutions 57
x
Chapter 3. TransformationGroups 61
1. A rolling triangle 61
x
2. Transformation groups 63
x
3
4 Contents
3. Classi(cid:12)cation of (cid:12)nite groups of movements 64
x
4. Conjugate transformations 66
x
5. Cyclic groups 70
x
6. Generators and relations 73
x
Chapter 4. Arbitrary groups 79
1. The general notion of a group 79
x
2. Isomorphism 85
x
3. The Lagrangetheorem 94
x
Chapter 5. Orbits and Ornaments 101
1. Homomorphism 101
x
2. Quotient group 104
x
3. Groups presented by generators and relations 107
x
4. Group actions and orbits 108
x
5. Enumeration of orbits 111
x
6. Invariants 117
x
7. Crystallographicgroups 118
x
Chapter 6. Other Types of Transformations 131
1. A(cid:14)ne transformations 131
x
2. Projective transformations 134
x
3. Similitudes 139
x
4. Inversions 144
x
5. Circular transformations 147
x
6. Hyperbolic geometry 150
x
Chapter 7. Symmetries of Di(cid:11)erential Equations 155
1. Ordinary di(cid:11)erential equations 155
x
2. Change of variables 158
x
3. The Bernoulli equation 160
x
4. Point transformations 163
x
5. One-parameter groups 168
x
6. Symmetries of di(cid:11)erential equations 170
x
7. Solving equations by symmetries 172
x
Answers, Hints and Solutions to Exercises 179
Preface 5
Preface
The (cid:12)rst Russian version of this book was written in 1983-1986 by B. D. Tcheb-
otarevsky and myself and published in 1988 by \Vysheishaya Shkola" (Minsk) under
the title \From ornaments to di(cid:11)erential equations". The pictures were drawn by
Vladimir Tsesler.
Years went by, and I was receiving positive opinions about the book from known
and unknown people. In 1996 I decided to translate the book into English. In the
course of this work I tried to make the book more consistent and self-contained. I
deletedsomeunimportantfragmentsandaddedseveralnewsections. Also,Icorrected
many mistakes (I can only hope I did not introduce new ones).
Thetranslationwasaccomplishedbythe year2000. In 2000,the Englishtextwas
further translated into Japanese and published by Springer Verlag Tokyo under the
title \Henkangun Nyu(cid:22)mon" (\Introduction to Transformation Groups").
The book is intended for high school students and university newcomers. Its aim
istointroducetheconceptof atransformationgrouponexamplesfromdi(cid:11)erentareas
ofmathematics. Inparticular,thebookincludesanelementaryexpositionofthebasic
ideas of S. Lie related to symmetry analysis of di(cid:11)erential equations that has not yet
appeared in popular literature.
The book contains a lot of exercises with hints and solutions. which will allow a
diligent reader to master the material.
The present version, updated in 2002, incorporates some new changes, including
the correction of errors and misprints kindly indicated by the Japanese translators
S. Yukita (Hosei University, Tokyo) and M. Nagura (Yokohama National University).
S. Duzhin
September 1, 2002
St. Petersburg
6 Contents
Introduction
Probably, the one most famous book in all history of mathematics is Euclid’s \Ele-
ments". InEuropeitwasusedasastandardtextbookofgeometryinallschoolsduring
about 2000years.
One of the (cid:12)rst theorems is the following Proposition I.5, of which we quote only
the (cid:12)rst half.
Theorem 1. (Euclid) In isosceles triangles the angles at the base are equal to one
another.
Proof. Every high school student knows the standard modern proof of this proposi-
tion. It is very short.
s
A
(cid:1)A
(cid:1) A
(cid:1) A
H(cid:1) (cid:8)A
(cid:1) A
(cid:1) A
(cid:1) A
B s(cid:1) AsC
Figure 1. Anisoscelestriangle
Standard proof. Let ABC be the given isosceles triangle (Fig.1). Since AB =
AC, thereexistsaplanemovement(re(cid:13)ection)thattakesAtoA,B toC andC toB.
Under this movement, \ABC goes into \ACB, therefore, these twoangles are equal.
It seems that there is nothing interesting about this theorem. However, wait a
little and look at Euclid’s original proof (Fig.2).
A
(cid:1)A
(cid:1)
rA
(cid:1)
A
(cid:1)
A
(cid:1)
A
(cid:1)
F(cid:1)!(cid:1)B(cid:1)!(cid:1)ar!a!a!a!a!!aAAarCAaAG
A
(cid:1) A
r r
D E
Figure 2. Euclid’sproof
Introduction 7
Euclid’s original proof. On the prolongations AD and AE of the sides AB
and AC choose two points F and G such that AF = AG. Then ABG = ACF,
4 4
hence \ABG = \ACF. Also CBG = BCF, hence \CBG = \BCF. Therefore
4 4
\ABC =\ABG \CBG=\ACF \BCF =\ACB.
(cid:0) (cid:0)
(cid:3)
InmediaevalEngland,PropositionI.5wasknownunderthenameofponsasinorum
(asses’ bridge). In fact, the part of Figure 2 formed by the points F, B, C, G and
the segments that join them, really resembles a bridge. Poor students who could not
master Euclid’s proof were compared to asses that could not surmount this bridge.
Figure 3. Asses’sBridge
From a modern viewpoint Euclid’s argument looks cumbersome and weird. In-
deed, why did he ever need these auxiliary triangles ABG and ACF? Why was not
he happy just with the triangle ABC itself? The reason is that Euclid just could
not use movements in geometry: this was forbidden by his philosophy stating that
\mathematical objects are alien to motion",
This example shows that the use of movements can elucidate geometrical facts
and greatly facilitate their proof. But movements are important not only if studied
separately. It is very interesting to study the social behaviour of movements, i.e. the
structureofsetsof movements (ormoregeneraltransformations)interrelatedbetween
themselves. Inthis area,the mostimportantnotion isthat of atransformation group.
Thetheoryofgroups,asamathematicaltheory,appearednotsolongago,onlyin
XIXcentury. However,examplesof objectsthataredirectlyrelatedtotransformation
groups,werecreatedalreadyinancientcivilizations,bothorientalandoccidental. This
referstotheartofornament,called\theoldestaspectofhighermathematicsexpressed
in an implicit form" by the famous XX century mathematician Hermann Weyl.
8 Contents
The following (cid:12)gure shows two examples of ornaments found on the walls of the
mediaeval Alhambra Palace in Spain.
a b
Figure 4. TwoornamentsfromAlhambra
Bothpatternsarehighlysymmetricinthesensethattherearepreservedbymany
plane movements. In fact, the symmetry properties of Figure 4a are very close to
those of Figure4b: eachornamenthasanin(cid:12)nite numberof translations,rotationsby
90(cid:14) and 180(cid:14), re(cid:13)ections and glide re(cid:13)ections. However, they are not identical. The
di(cid:11)erencebetweenthemisinthewaythesemovementsarerelatedbetweenthemselves
for eachof the two patterns. The exactmeaning of these wordscanonly be explained
in terms of group theory which says that symmetry groups of (cid:12)gures 4a and 4b are
not isomorphic (this is the contents of Exercise 129, see page 129).
Theproblemtodetermineandclassifyallthepossibletypesofwallpatternsymme-
try was solved in late XIX century independently by a Russian scientist E.S.Fedorov
andaGermanscientistG.Scho(cid:127)n(cid:13)iess. Itturnedoutthatthereareexactly17di(cid:11)erent
types of plane crystallographic groups (see the table on page 126).
Of course, signi(cid:12)cance of group theory goes far beyond the classi(cid:12)cation of plane
ornaments. In fact, it is one of the key notions in the whole of mathematics, widely
used in algebra, geometry, topology, calculus, mechanics etc.
This book provides an elementary introduction into the theory of groups. We be-
gin with someexamples fromelementaryEuclideangeometrywhereplanemovements
play an important role and the ideas of group theory naturally arise. Then we ex-
plicitly introduce the notion of a transformation group and the more general notion
of an abstract group, discuss the algebraic aspects of group theory and its applica-
tions in number theory. After this we pass to group actions, orbits, invariants, some
classi(cid:12)cationproblems and (cid:12)nally goas far as the applicationof continuousgroupsto
the solution of di(cid:11)erential equations. Our primary aim is to show how the notion of
group works in di(cid:11)erent areas of mathematics thus demonstrating that mathematics
is a uni(cid:12)ed science.
Introduction 9
The bookisintendedforpeoplewith highschoolmathematicaleducation, includ-
ing the knowledge of elementary algebra, geometry and calculus.
Youwill(cid:12)ndmanyproblemsgivenwith detailedsolutionsandlotsofexercisesfor
self-study supplied with hints and answers at the end of the book. It goes without
saying that the reader who wants to really understand what’s going on, must try to
solve as many problems as possible.