The Theory of Finite Groups: An Introduction (Universitext)

I don't think I could come up with enough superlatives to describe just how good this text really is. Both authors have done an excellent job describing as well as motivating the subject and have subsequently turned me into an avid fan of the theory of finite groups. What I find disappointing is my assumption that the available pool of readers is more like a shallow puddle. This can easily be seen from the fact that, as thriving of a subject finite groups may be, it does not appear in the course catalog of most academic institutions. It would be nice to see this book adopted as a seniors honor course for undergraduates or as an advanced algebra course for graduate students looking to specialize in matters algebraic. Now, off my soap box. Without a doubt this text will become another classic in the field of finite groups, akin to Gorenstein's. So who can read this text? Well, as un-original as it is, the mathematically mature and, more importantly, anyone who has taken an Algebra course that covered the standard basics of group theory. To be exact, if you learned Algebra using Hungerford's text, then you should be adequately prepared for this book. Of course this assumes you have seen and understand the how's/why's of the Sylow theorems, the structure and purpose of Abelian groups, the role Normality plays and, in my opinion, you should have a firm grasp of the role/structure/purpose of the various types of morphisms (homomorphism, epimorphism, monomorphism, and isomorphism)and how and why they are so important in group theory. The only drawback to this book is that the exercises seemingly disappear after page 223 in the chapter dealing with groups acting on groups. I know the material becomes more dense, more difficult and more abstract but, I would have assumed that the authors could have provided the reader with exercises post chapter 8 (even if the exercise was finishing a proof). Aside from that, I would recommend this text to everyone and anyone. In particular, if you are not a math wiz and just plain interested in learning about finite groups, then I would suggest that you pick up this text. Good luck and happy proving!

From reviews of the German edition:

This is an exciting text and a refreshing contribution to an area in which challenges continue to flourish and to captivate the viewer. Even though representation theory and constructions of simple groups have been omitted, the text serves as a springboard for deeper study in many directions. One who completes this text not only gains an appreciation of both the depth and the breadth of the theory of finite groups, but also witnesses the evolutionary development of concepts that form a basis for current investigations. This is accomplished by providing a thread that permits a natural flow from one concept to another rather than compartmentalizing. Operators on sets and groups are introduced early and used effectively throughout. The bibliography provides excellent supplemental support...The text is tight; there is no fluff. The format builds on concepts essential for later expansion and associated reading. On occasion, results are stated without proof; continuity is maintained. Several proofs are provided free of representation theory on which the originals were based. More generally the proofs are direct, perhaps at times brief. The focus is on the underlying structural components, with selected details left to the reader. As a result the reader develops the maturity required for approaching the literature with confidence. The first eight chapters have an abundance of exercises, not prorated, and some of the more challenging are addressed later in the text. Due to the nature of the material, fewer exercises appear in the remaining chapters. (H. Bechtell, Mathematical Reviews)

From reviews of the German Edition: "This is an exciting text and a refreshing contribution to an area in which challenges continue to flourish and to captivate the viewer. Even though representation theory and constructions of simple groups have been omitted, the text serves as a springboard for deeper study in many directions. One who completes this text not only gains an appreciation of both the depth and the breadth of the theory of finite groups, but also witnesses the evolutionary development of concepts that form a basis for current investigations. This is accomplished by providing a thread that permits a natural flow from one concept to another rather than compartmentalizing. Operators on sets and groups are introduced early and used effectively throughout. The bibliography provides excellent supplemental support." H. Bechtell, Mathematical Reviews "Altogether, we have a well written book, which gives an introduction into the field and furthermore shows us one of the most active recent areas of research. This is the first book which shows us the amalgam method and moreover shows us how it works. Maybe it can get the same influence on group theory today as Gorenstein's famous book got in the late sixtees and seventees. This book should be in any library." G. Stroth, Zentralblatt