Cover 1
Title 4
Copyright 5
Contents 6
Preface 10
Introduction 12
Chapter 1. Algebra of Points 18
§1. Checkered plane 18
§2. Point addition 21
§3. Multiplying points by numbers 25
§4. Centre of gravity 28
§5. Coordinates 31
§6. Point multiplication 35
§7. Complex numbers 41
Chapter 2. Plane Movements 52
§1. Parallel translations 52
§2. Reflections 55
§3. Rotations 58
§4. Functions of a complex variable 61
§5. Composition of movements 66
§6. Glide reflections 72
§7. Classification of movements 74
§8. Orientation 77
§9. Calculus of involutions 79
Chapter 3. Transformation Groups 84
§1. A rolling triangle 84
§2. Transformation groups 87
§3. Classification of finite groups of movements 89
§4. Conjugate transformations 91
§5. Cyclic groups 97
§6. Generators and relations 101
Chapter 4. Arbitrary Groups 108
§1. The general notion of a group 108
§2. Isomorphism 117
§3. The Lagrange theorem 129
Chapter 5. Orbits and Ornaments 138
§1. Homomorphism 138
§2. Quotient group 142
§3. Groups presented by generators and relations 147
§4. Group actions and orbits 148
§5. Enumeration of orbits 152
§6. Invariants 159
§7. Crystallographic groups 162
Chapter 6. Other Types of Transformations 176
§1. Affine transformations 176
§2. Projective transformations 180
§3. Similitudes 186
§4. Inversions 193
§5. Circular transformations 198
§6. Hyperbolic geometry 202
Chapter 7. Symmetries of Differential Equations 208
§1. Ordinary differential equations 208
§2. Change of variables 213
§3. The Bernoulli equation 214
§4. Point transformations 218
§5. One- parameter groups 225
§6. Symmetries of differential equations 227
§7. Solving equations by symmetries 231
Answers, Hints and Solutions to Exercises 240
Index 256
A 256
B 256
C 256
D 256
E 256
F 256
G 256
H 257
I 257
K 257
L 257
M 257
N 257
O 257
P 257
Q 257
R 257
S 257
T 257
V 257
W 257
Back Cover 258
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