2012 • 192 Pages • 1.17 MB • English

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

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Contents Preface 5 Introduction 6 Chapter 1. Algebra of points 11 x1. Checkered plane 11 x2. Point addition 13 x3. Multiplying points by numbers 17 x4. Centre of gravity 19 x5. Coordinates 21 x6. Point multiplication 24 x7. Complex numbers 28 Chapter 2. Plane Movements 37 x1. Parallel translations 37 x2. Re�ections 39 x3. Rotations 41 x4. Functions of a complex variable 44 x5. Composition of movements 47 x6. Glide re�ections 52 x7. Classi�cation of movements 53 x8. Orientation 56 x9. Calculus of involutions 57 Chapter 3. Transformation Groups 61 x1. A rolling triangle 61 x2. Transformation groups 63 3

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4 Contents x3. Classi�cation of �nite groups of movements 64 x4. Conjugate transformations 66 x5. Cyclic groups 70 x6. Generators and relations 73 Chapter 4. Arbitrary groups 79 x1. The general notion of a group 79 x2. Isomorphism 85 x3. The Lagrange theorem 94 Chapter 5. Orbits and Ornaments 101 x1. Homomorphism 101 x2. Quotient group 104 x3. Groups presented by generators and relations 107 x4. Group actions and orbits 108 x5. Enumeration of orbits 111 x6. Invariants 117 x7. Crystallographic groups 118 Chapter 6. Other Types of Transformations 131 x1. A�ne transformations 131 x2. Projective transformations 134 x3. Similitudes 139 x4. Inversions 144 x5. Circular transformations 147 x6. Hyperbolic geometry 150 Chapter 7. Symmetries of Di�erential Equations 155 x1. Ordinary di�erential equations 155 x2. Change of variables 158 x3. The Bernoulli equation 160 x4. Point transformations 163 x5. One-parameter groups 168 x6. Symmetries of di�erential equations 170 x7. Solving equations by symmetries 172 Answers, Hints and Solutions to Exercises 179

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Preface 5 Preface The �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�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 deleted some unimportant fragments and added several new sections. Also, I corrected many mistakes (I can only hope I did not introduce new ones). The translation was accomplished by the year 2000. In 2000, the English text was further translated into Japanese and published by Springer Verlag Tokyo under the title \Henkangun Nyu�mon" (\Introduction to Transformation Groups"). The book is intended for high school students and university newcomers. Its aim is to introduce the concept of a transformation group on examples from di�erent areas of mathematics. In particular, the book includes an elementary exposition of the basic ideas of S. Lie related to symmetry analysis of di�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

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6 Contents Introduction Probably, the one most famous book in all history of mathematics is Euclid’s \Ele- ments". In Europe it was used as a standard textbook of geometry in all schools during about 2000 years. One of the �rst theorems is the following Proposition I.5, of which we quote only the �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 �A � A � A � A H � � A � A � A B s� AsC Figure 1. An isosceles triangle Standard proof. Let ABC be the given isosceles triangle (Fig.1). Since AB = AC, there exists a plane movement (re�ection) that takes A to A, B to C and C to B. Under this movement, \ABC goes into \ACB, therefore, these two angles 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). r A �A � A � A � A � A � B r ArC F r!��!�a!a!a!a!a!!aAaAaArG � A � A D E Figure 2. Euclid’s proof

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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 4ABG = 4ACF, hence \ABG = \ACF . Also 4CBG = 4BCF, hence \CBG = \BCF . Therefore \ABC = \ABG�\CBG = \ACF �\BCF = \ACB. � In mediaeval England, Proposition I.5 was known under the name of pons asinorum (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’s Bridge 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 structure of sets of movements (or more general transformations) interrelated between themselves. In this area, the most important notion is that of a transformation group. The theory of groups, as a mathematical theory, appeared not so long ago, only in XIX century. However, examples of objects that are directly related to transformation groups, were created already in ancient civilizations, both oriental and occidental. This refers to the art of ornament, called \the oldest aspect of higher mathematics expressed in an implicit form" by the famous XX century mathematician Hermann Weyl.

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8 Contents The following �gure shows two examples of ornaments found on the walls of the mediaeval Alhambra Palace in Spain. a b Figure 4. Two ornaments from Alhambra Both patterns are highly symmetric in the sense that there are preserved by many plane movements. In fact, the symmetry properties of Figure 4a are very close to those of Figure 4b: each ornament has an in�nite number of translations, rotations by � � 90 and 180 , re�ections and glide re�ections. However, they are not identical. The di�erence between them is in the way these movements are related between themselves for each of the two patterns. The exact meaning of these words can only be explained in terms of group theory which says that symmetry groups of �gures 4a and 4b are not isomorphic (this is the contents of Exercise 129, see page 129). The problem to determine and classify all the possible types of wall pattern symme- try was solved in late XIX century independently by a Russian scientist E. S. Fedorov and a German scientist G. Scho�n�iess. It turned out that there are exactly 17 di�erent types of plane crystallographic groups (see the table on page 126). Of course, signi�cance of group theory goes far beyond the classi�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 some examples from elementary Euclidean geometry where plane movements 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�cation problems and �nally go as far as the application of continuous groups to the solution of di�erential equations. Our primary aim is to show how the notion of group works in di�erent areas of mathematics thus demonstrating that mathematics is a uni�ed science.

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Introduction 9 The book is intended for people with high school mathematical education, includ- ing the knowledge of elementary algebra, geometry and calculus. You will �nd many problems given with detailed solutions and lots of exercises for 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.

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