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Finite Group Theory

I. Martin Isaacs
American Mathematical Society
Publication Date: 
Number of Pages: 
Graduate Studies in Mathematics 92
BLL Rating: 

The Basic Library List Committee recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Andrew Locascio
, on

The late Gian-Carlo Rota once wrote a paper entitled “Ten Things I Wish I Had Been Taught.” Like most of Rota’s writing, it’s a model of clarity and wisdom and is well worth chasing down. (It is included in his Indiscrete Thoughts.) The fourth item on Rota’s list is that “You are most likely to be remembered for your expository works.” It’s easy to see why this is true: no matter how prolific a researcher is, the overwhelming majority of his or her papers will be incomprehensible to all but specialists. His textbooks, on the other hand, can reach a much larger audience.

The fact that an expert’s true legacy, when all is said and done, will probably be his or her textbooks may be the single most tragic irony of today’s “publish or perish” mentality. There are more then a few examples in the recent history of mathematics alone. S. S. Chern revolutionized differential geometry with the use of moving frames, the invention of characteristic classes, the modern concept of a connection and so much more, but he’ll probably always be most remembered for the yellowing University of Chicago mimeographed lecture notes from the 1950s. An entire generation of geometers learned the elements of differentiable manifolds from those notes. John Milnor is probably the foremost topologist of the second half of the 20th century, but many of us first learned about him by reading his book on differential topology. And of course, Milnor’s most famous student by far is one who published a bare handful of forgotten papers in the 30+ years since receiving his doctorate — yet arguably has had an even greater long term impact on mathematics as a whole through his writings than Milnor himself: Michael Spivak.

Writing advanced textbooks — those for budding young mathematician in graduate school — is a difficult but important undertaking: these works speak directly to those from whom the next generation of discoveries will come. After all, any schmuck with a BA, TeX, and some computer graphic programming knowhow can write a calculus text. But how many of us can write a comprehensible text on elliptic curves or complex manifolds? That takes someone who not only knows large chunks of the overall structure of modern mathematics intimately, but also has thought a great deal about the best ways to share that knowledge with others.

Which brings me to this handsome blue volume in the AMS’s Graduate Studies in Mathematics series. I. Martin Isaacs is a veteran algebraist of some renown. He was a student of Richard Brauer at Harvard and has produced some beautiful results on groups and representation theory over the years. He has also won a number of teaching awards, which should come as no surprise to those of us who have read his wonderful textbooks. His first, Character Theory of Finite Groups, has been reprinted in the AMS/Chelsea series and is one of the standard texts on the subject. The text Isaacs is probably most famous for is his Algebra: A Graduate Course, out of print for a number of years but recently reissued by the AMS. It’s an amazing book that covers basic algebra in a beautifully written, comprehensive and strikingly original manner. As this is one of my all time favorite textbooks, I jumped at the chance to review his book on finite group theory. He did not disappoint.

Finite group theory is probably the oldest branch of modern algebra. As such, it’s a subject with no shortage of good textbooks for a graduate course. So when one sits down to review a new one, the first question you ask yourself is “what’s different about this one? What does this one have that the others don’t?”

Well, for one thing, it’s a lot more advanced then the usual group theory text. Most books on group theory assume very little background of the reader. This is presumably to increase the number of potential readers, though most students taking a graduate course in group theory will have had at least an honors course in algebra à la Herstein’s Topics In Algebra or Artin’s Algebra. Perhaps the problem is that the background knowledge is going to vary enormously from program to program. So many books on group theory assume minimal background. Not so with this book. There is an appendix at the end entitled “The Basics,” but it’s clear that anyone without a good working knowledge of modern algebra needs to obtain such knowledge before reading this book. The advantage is that Isaacs can go much deeper into a lot of aspects of finite groups. I say bully for him. Graduate students taking group theory who don’t know what a factor group is should be shown the door and told stop wasting everyone else’s time.

Isaacs’ first chapter gives a long account of the Sylow theorems beginning with Wielandt’s proof based on group actions. The central concept of the book is indeed that of a group action — as well it should be in a modern group theory text. Isaacs gives a fairly complete overview of the “decomposition theory” of finite groups and the importance of the Sylow analysis as a tool in deriving these results. Along the way, we meet several results that I doubt have ever appeared in book form before, such as Brodkey’s Theorem and the Chernak-Delgado measure.

Chapter 2 is about an important topic to that, again, to my knowledge, has never appeared in a textbook: subnormality. It is defined as follows: A subgroup S of G is said to be subnormal iff there exists a finite chain of subgroups Hi such that S = H0 ⊳  H1 ⊳ … ⊳ Hr = G, where r ∈ N. Subnormality is a transitive relation on the set of subgroups of G, unlike normality. It turns out that a great deal of the structure of a finite group G can be analyzed using this property. For example, many of the properties of nilpotent groups follow easily from those of their subnormal subgroups, since a group G is nilpotent iff all of its subgroups are subnormal. This is only a taste of the very rich treasures presented in this chapter, including the Wielandt subgroup, the “zipper lemma” (for all subgroups of a finite group G where S, H are proper subgroups, S is subnormal and S ⊂ H, there exists a unique subgroup J which is maximal and S ⊂ J) and the deep relations between subnormal and solvable subgroups of G.

The following chapters present more standard material for a first course in group theory, although again presented in original ways and interspersed with new or relatively obscure results appearing in book form for the first time. Chapter 3 presents the theory of group extensions, so important in Galois Theory and homology. Isaacs motivates the theory with a lucid account of the ambiguities surrounding group extension constructions and why it’s so critical to have a method that completely determines a given extension group G from a chosen conjugation action on a subset of ordered pairs of H x N where Ω  is a nonempty set and H and N are subgroups of Sym (Ω). This discussion is quite enlightening and was one of my favorite parts of the text. This is not an easy subject to motivate; very few authors even try. Isaacs does a great job in just 3 pages. This immediately gives an indirect yet completely general construction of the semidirect product. Using this approach, Isaacs goes on to discuss split extensions, the wreath product of H with G and its base group B, crossed homomorphisms, transversals, and a very complete, step by step proof of the Schur-Zassenhaus theorem.

Chapter 4 discusses the commutator subgroups of finite groups and their relation to automorphisms, conjugacy and nilpotency classes, culminating in Thompson’s P x Q Theorem and a discussion of its importance in the classification of the finite simple groups. Chapter 5 is on the transfer homomorphism, so critical in character theory. It is here that Isaacs begins to develop a foundation for further courses in finite group representation theory, one of his stated purposes. Beginning by defining the right transversal of a finite group G, he defines the pretransfer and transfer maps, showing these functions are defined independent of the choice of right transversal of G. He then proceeds in rapid fire succession to construct the essential machinery of transfer theory: perfect groups, the Schur multiplier of a group and the Schur representation group for G. He then proves the two big theorems of transfer theory: the Burnside p-complement theorem and the Frobenius Theorem on the characterization of normal p-complements of p-subgroups.

Chapter 6 describes Frobenius actions on finite groups and their regular and semiregular orbits. Frobenius groups, their properties and their applications to matrix groups are described. This is to lay the groundwork for chapter 7 on the Thompson characteristic subgroup of finite groups and the resulting 1964 result by Thompson proving all Frobenius group kernels are nilpotent. Along the way, Isaacs discusses, in illuminating digressions, the grand saga of the classification of the finite simple groups and the pivotal role this result of Thompson played.

Chapter 8 is on permutation groups. Most of this material is standard: permutation isomorphisms, regular and transitive actions the various kinds of transitivity among actions on such groups. The chapter culminates in the Jordan transposition theorem. At the end of this chapter, there are some exciting applications to graph theory and geometry, including a group action proof of the simplicity of the projective special linear group PSL(n,R). Chapter 9 gives more results on subnormal subgroups, building on chapter 2. The book concludes with chapter 10, on further topics in transfer theory, including Yoshida’s generalization of the Hall-Weilandt theorem on normalizer control of p-transfer, Mashke’s and Huppert’s theorems in subgroup automorphisms and their applications. This concluding material is quite difficult and really should be of use mostly to graduate students looking to specialize in algebra.

As you can see from the description, this is one serious group theory book, intended for graduate students with strong algebra backgrounds who plan to read papers on group theory after this course. It certainly belongs in the AMS Graduate Studies series. Each chapter comes with a freight car full of substantial exercises, ranging in difficulty from trivial to research level, many of them defining aspects of group theory not covered in the text proper, such as the Frattini subgroup, elementary abelian groups, the quasiquaternion and generalized quaternion groups, extraspecial groups, supersolvable groups and much, much more — some of which are later used in the text proper. The book also has the one telling characteristic of a text written by an active researcher in the field — the material covered reaches much closer to the research frontier then is usual. This is particularly clear in the chapters on subnormality and transfer theory, which contain many fairly recent results.

The book is amazingly clean. I couldn’t find a single error. But the very best thing Isaacs brings to this book is the same thing he brings to all his textbooks — his wonderful style. Definitions, theorems and associated results are presented in a remarkably well organized, coherent manner, all in the author’s terrific lively prose. In this regard, the book reads at times less like a textbook and more like a novel on the great narrative of the story of the development of finite group theory over the last twelve decades. The running theme unifying all these results in the narrative is the great accomplishment of the classification of finite simple groups. The flowing, eclectic style certainly conveys the vast love of the author for his chosen specialty and his great desire to set others on the same path.

I almost hate to bring it up, but I do have one tiny quibble with the book — and I’d be remiss as a reviewer if I didn’t bring it up. Except for the first chapter, there are almost no specific examples of finite groups presented, even in the exercises! This is almost a trademark of Isaacs’ earlier books, and frankly, it makes me nuts. He almost never gives specific examples of things, despite the thoroughness and completeness with which he presents material. In learning advanced mathematics, a few good, detailed examples are worth more than a hundred complete proofs. With advanced material, this becomes even more important.

Still, this is really a matter of personal taste rather then an actual criticism of the book. Isaacs has written yet another masterful text and I have no doubt it will soon join Marshall Hall’s The Theory of Groups, Joseph J. Rotman’s An Introduction to the Theory of Groups and John Rose’s A Course In Group Theory as a classic text in modern group theory and the textbook of choice for the very best students of graduate algebra. If you’re studying group theory anytime soon, get a copy. You’ll thank me later, I promise.

Andrew Locascio is entering his final semester of master’s work at Queens College of the City University of New York.