Introduction to Group Theory
There are many introductions to group theory. This one tries to be different. Firstly, it skips a lot of standard material to make space for several dives into quite advanced topics and, secondly, it gives preference to combinatorial group theory over finite and matrix groups. The book is divided into three parts of uneven length. The first part starts with basic notions followed by Sylow’s theorem. The author then starts to discuss finite simple groups. He begins with automorphisms of the icosahedron, proves the simplicity of A5 and then moves to a representation of the Mathieu group M22 by automorphisms of ovals and lines in an extension of P2(4). This is followed by a brief discussion of extensions and by a description of the Higman-Sims group. The first part ends with a diagram of sporadic groups. The rest of the book is concerned with combinatorial group theory. The second part presents most of the standard topics (Tietze transformations, Reidemeister-Schreier, amalgamated free products, HNN, Kurosh, van Kampen, coverings and Hopfian groups). The exposition relies upon an efficient formalism that may be a bit surprising for the beginner. The third part is aimed at proving the theorem of Bestvina and Handel that describes irreducible outer automorphisms of a free group by means of a train track map. This part relies heavily on the Perron-Frobenius eigenvalue. There are some (but not many) exercises in the book and detailed solutions are provided for more than twenty of them.