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Introduction to Group Theory

Oleg Bogopolski
European Mathematical Society/ American Mathematical Society
Publication Date: 
Number of Pages: 
EMS Textbooks in Mathematics
[Reviewed by
Luiz Henrique de Figueiredo
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This short book aims to introduce the reader to mainstream topics in group theory: sporadic groups and the classification of finite simple groups; combinatorial and geometric group theory; and the theory of train tracks for automorphisms of free groups.

The first chapter contains a quick-paced introduction to finite group theory, including all basic theorems. The flavor is geometric: for instance, the proof of Sylow's theorem uses embeddings into general linear groups. The second half of the chapter focuses on finite simple groups and includes a detailed geometric interpretation of three important examples: the alternate group A5, the Mathieu group M22, and the Higman­-Sims group HS.

The second chapter is devoted to combinatorial group theory, with emphasis on the action of groups on graphs. especially the Bass­-Serre theory of groups acting on trees. The chapter starts the standard material on free groups, presentations by generators and relations, Tietze transformations, HNN extensions, and then moves on to more advanced material such as the Seifert-van Kampen theorem.

The third and last chapter contains an exposition of the recent Bestvina-Handel theory of train tracks for automorphisms of free groups. An appendix on the Perron­-Frobenius theorem complements this chapter.

Although the author says in the preface that "the reader is assumed to have the knowledge of algebra expected after the first semester of university (permutations, fields, matrices, vector spaces)", I think the book is best suited for graduate students because the exposition is quick-paced throughout the book.

The book can be used for a graduate course in group theory, but it can also be read independently. In both cases, the student that manages to work through the book in detail will be rewarded with a working knowledge of advanced, mainstream topics in group theory.

Luiz Henrique de Figueiredo is a researcher at IMPA in Rio de Janeiro, Brazil. His main interests are numerical methods in computer graphics, but he remains an algebraist at heart. He is also one of the designers of the Lua language.

  • Introduction to finite group theory
  • Introduction to combinatorial group theory
  • Automorphisms of free groups and train tracks
  • Appendix. The Perron-Frobenius Theorem
  • Solutions to selected exercises
  • Bibliography
  • Index