This is the third edition of the book which earlier appeared under the title “The theory of groups: an introduction” (1st ed. 1965, 2nd ed. 1973). For a review of the first edition, see this Zbl 0123.020. The slight change in the title signals a considerable reorganization of the material: the order of some topics was changed, new material has been added including a whole new chapter on permutations and the Mathieu groups, many sections were rewritten, exercises added.
The book starts from the very beginnings of group theory, assuming only some knowledge of linear algebra and rudiments of rings and fields. Several topics which have close connections with group theory are shortly introduced, such as some Galois theory, affine and projective geometry, Steiner systems, some homological algebra. Numerous exercises serve to make the results understood and also to develop some new theory. All this has created a fresh new version of an excellent textbook for students who want to become acquainted on a nontrivial level with group theory. Here are the chapter titles for a better idea of the contents: Groups and Homomorphisms, The Isomorphism Theorems, Permutation Groups, The Sylow Theorems, Normal Series, Finite Direct Products, Extensions, Some Simple Linear Groups, Permutations and the Mathieu Groups, Infinite Abelian Groups, Homological Algebra, Free Groups and Free Products, The Word Problem.