This is a self-contained introduction to the theory of finite groups. The treatment is exhaustive, from the elementary basic results up to characters and representations of finite groups, with applications to Burnside’s paqb-theorem and Frobenius kernels. The main results are accompanied by several well-chosen examples, and there are many computations with groups of small order, giving the reader a sense of concreteness.
The topics covered include: Basic definitions, subgroups, cosets, Lagrange’s theorem, normal subgroups, permutation and matrix groups, homomorphisms, Noether isomorphism theorems, actions and the orbit-stabilizer theorem, p-groups, and Sylow theory, direct products and finite abelian groups, semi-direct products, composition series and the Jordan-Hölder theorem, nilpotent and solvable groups, Frattini and Fitting subgroups, Hall’s generalization of Sylow’s theorems, some simple groups (projective special lineal groups over finite fields). Although in chapter 12 there is some discussion of Steiner systems and the Mathieu groups, essentially only two Steiner systems are constructed, S(5,6,12) and S(4,5,11), and the corresponding Mathieu groups M11 and M12 are discussed in detail.
One novelty of this book is that additional sections for Chapters 3 to 12 plus two extra chapters on (complex) characters and (linear) representations of finite groups, which include applications to Burnside’s paqb-theorem and Frobenius kernels, are being put on the book website. At the time when this review was written, the web sections already uploaded included a solution appendix, the extra sections of chapters 3 and 12, and the two extra chapters 13 and 14. According to the table of contents in the print version of the book, there should be also new sections on chapters 4, 5, 6, 7 and 9, with important topics such as the transfer homomorphism, the Schur-Zassenhaus theorem, Burnside’s normal complement theorem and Iwasawa’s lemma to test simplicity, which is used to give an alternate proof of the simplicity of the projective linear groups, for example.
This is a well-written book, not too wordy nor too terse. Concepts and results are illustrated with examples, and the problem sets at the end of every chapter nicely complement the theory. There are few books at the same level as the book under review, addressed to advanced undergraduates or beginning graduate students. One that comes to mind is Rotman’s An Introduction to the Theory of Groups Rotman has more on general groups, presentation theory and the Mathieu groups, but has nothing on characters and representations, hence when discussing Hall’s theorems or Burnside’s, Rotman just quotes the corresponding results, which are given full proofs in Rose’s book. Having said this, it is clear that I am keeping both books (I have Rotman’s Allyn and Bacon 2nd edition of 1978, with a permutation of the title words), since both complement each other nicely.
After reading this book, the student will be well prepared to more advanced books such as Isaac’s Finite Group Theory (AMS, 2008), or Kurzweil and Stellmacher The Theory of Finite Groups (Springer, 2004) to mention just two recent additions to a large literature, or more classical texts such as Hall’s The Theory of Groups (AMS-Chelsea, 1999), or Gorenstein’s Finite Groups (AMS-Chelsea, 2007).
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City, and for full disclosure would like to add that, under the spell of group theory, he also has felt to the temptation to publish a textbook on finite groups, now in its second printing (SMM-Reverté, 2006). His e-mail address is firstname.lastname@example.org.