This is a slow paced introduction to transformation groups intended for undergraduate students. The original version was published in Russian in 1988 under the title From Ornaments to Differential Equations. For the English translation the original text was revisited and expanded. The book is divided into seven chapters. The first one introduces coordinate systems in the plane and the algebraic operations on points of the plane. The second one introduces various elementary transformations of the plane. The third chapter brings the concept of a group of transformations of the plane including further concepts like conjugated transformations or generators and relations for transformation groups. The fourth chapter is about abstract groups and the Lagrange theorem. The fifth chapter applies results of the previous chapters to the classification of ornaments and to crystallographic groups. Chapter 6 introduces further types of transformations in the plane like projective transformations, inversions and hyperbolic transformations. The final chapter is about symmetries of differential equations and how they can be used to solve the equations. The book is well written and it contains a lot of exercises with hints and solutions.

Reviewer:

jtu