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Fundamentals of Group Theory: An Advanced Approach

Steven Roman
Publication Date: 
Number of Pages: 
[Reviewed by
Mark Hunacek
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A careful look at the title of this book reveals that the word “advanced” modifies “approach” rather than “group theory,” which in turn is described as “fundamental”. This is truth in advertising: the topics covered in this book are, for the most part, quite fundamental; the book assumes no prior knowledge of group theory, begins with the definition of “group,” and covers topics which are, largely, things which a person with a good undergraduate abstract algebra course under his or her belt would likely have been exposed to, though probably not at this level of sophistication.

Some topics that one might consider fairly basic are not covered until pretty late in the text. Sylow theory, for example, does not appear until about two-thirds of the way in. In this respect, this book should be distinguished from texts such as Isaacs’ Finite Group Theory, accurately described by Andrew Locascio in his MAA review as being “one serious group theory book” (emphasis in the original) and which, assuming a standard first-year graduate algebra course, begins with a chapter on group actions and Sylow theory. This book, by contrast, is not intended for second-year (or later) graduate students wanting to learn a lot of sophisticated group theory; it is instead a text for beginning graduate students taking an introductory group theory course or, perhaps more likely given most graduate syllabi, a first-year course in abstract algebra in which all or most of a semester is devoted to groups. As a result, this text omits the proof of some significant results in group theory, such as Burnside’s paqb theorem, which is proved in Isaacs’ book.

The first six chapters of this text, comprising a little more than half the book, discuss topics the names of which would, for the most part, likely be familiar to students in a first course in abstract algebra: groups, homomorphisms and isomorphisms, cyclic groups, normal subgroups and quotient groups, permutation groups, direct products and the fundamental theorem of finite abelian groups, etc. The difference between the discussion here and in most introductory texts lies in the succinctness of the exposition and the level of detail of some of the discussions. Some topics (such as subnormal series and semidirect product) may not be addressed at all in a typical introductory book and others (such as commutators) that may be are treated here in much more detail here than one usually sees. The author gives, for example, an explicit example of a group G where the commutator subgroup G’ does not consist solely of simple commutators xyx-1y-1. I had seen this particular example only once before, in an exercise in the third edition of Rotman’s An Introduction to the Theory of Groups. There is also a nice discussion here of how Galois viewed the concept of a group; the material in this short section was new to me. Also new to me was the concept of aC, nC, aD, and nD groups; these are groups that satisfy certain properties regarding whether general subgroups and normal subgroups are complemented or direct sums. A number of results concerning these classes of groups are given, some without proof (but with references).

There were a few times where it seemed to me the author made things a bit more confusing than they needed to be. As an illustration, consider his discussion of cyclic groups. On page 22 of the text, the author begins by considering a to be a “formal symbol” and then defining a group (“the cyclic group generated by a”) consisting of all formal integral powers of a, with group operation defined by the usual law of exponents. He separately defines “the cyclic group of order n” to be the set of formal integral powers of a with product defined by adding exponents modulo n, and immediately gives as an example the usual cyclic group Zn, which he claims to be generated by 1. Apart from the potentially confusing idea of considering the element 1 in Zn ,which already has an independent meaning, as a “formal symbol”, these definitions seem to suggest that the “cyclic group of order n” is not generated by anything, since the term “cyclic group generated by a” is defined only for infinite groups. It is not until about ten pages later (page 34) that the author then defines the “cyclic subgroup generated by a” (a now being an element in a group G) in the usual manner and then another ten pages later makes reference to general “cyclic groups” without, so far as I can see, ever actually defining a group to be “cyclic” if it is equal to a cyclic subgroup generated by some element.

The final seven chapters of the book cover topics that are generally discussed in a first-year graduate algebra course, and some, perhaps, in an occasional advanced undergraduate course. These include group actions, Sylow theory, solvable and nilpotent groups, free groups and presentations, and finitely-generated abelian groups. Here again there are some interesting tidbits. Chapter 9, for example, is a nice overview of the classification theory of finite simple groups, which gives at least some sense of how this massive undertaking was approached and (probably, anyway) solved. (The author states that some believe it is “too soon to state categorically” that the problem has been solved but is “generally believed by experts in the field” to have been.) Another nice feature to be found here, and not easily located elsewhere in the textbook literature, is the discussion in chapter 8 of the Sylow subgroups of the symmetric group Sn. (The classification of these involves wreath products and a preliminary reduction to the case where n is a prime power.)

It is these occasional tidbits that, I think, constitute the primary value of this book: just about anyone reading this review already knows that it’s always nice to have at one’s fingertips a source for hard-to-find theorems and counter-examples. I do wish, however, that the author had provided a chapter on basic group representation theory, which, as has been demonstrated in books such as Steinberg’s Representation Theory of Finite Groups, can be addressed without extensive background in ring or module theory.

I would also have liked to have seen more extensive discussion of interesting examples, particularly matrix groups: although the linear and special linear groups are defined as examples early in the book, not much is done with them, and the projective special linear groups — which, over finite fields, give interesting examples of simple groups — are not mentioned. Also not mentioned, despite the fact that the author explicitly discusses systems of distinct representatives for both left and right cosets, is the interesting result of Hall (a consequence of his “marriage theorem”) that for any subgroup H of a finite group G, there exists a set which is a system of distinct representatives for both the left and right cosets of H.

These mild quibbles notwithstanding, there is much to recommend in this book for a student taking an introductory graduate course in group theory or an instructor teaching such a course. Careful study of this book will provide a solid grounding in the foundations of the subject and certainly prepare a student for the study of more advanced topics.

Mark Hunacek ( teaches mathematics at Iowa State University.