A First Course in Group Theory

This book was written as a textbook for a first course in group theory. Starting from first principles, the book slowly builds to the isomorphism theorems, covering a number of interesting special topics along the way.

 

Many introductory group theory textbooks have such a heavy emphasis on finite groups that students leave them not really fully aware that there are infinite groups other than the additive groups of integers, real numbers, and complex numbers. Davvaz's book, on the other hand, features many excellent discussions of groups of matrices. Indeed, matrix groups are used not just as examples of groups, but to help clarify and add depth to Davvaz's discussion of other families of groups. For example, in the chapter on matrix groups, Davvaz uses the orthogonal and special orthogonal groups to provide a more nuanced treatment of the isometries of the plane, a topic covered at the beginning of the book. The chapter on matrix groups is by far the longest of the text (it accounts for nearly twenty percent of the book's length). It is also, in my opinion, the highlight of the book.

 

Davvaz's book does not cover nearly as much material as other texts. For example, the structure theorem for finite abelian groups, group actions, and the Sylow theorems all go unmentioned. In fact, isomorphisms are only defined twenty pages before the end of the book. Instead, Davvaz covers a wide variety of applications in much greater depth than is customary in introductory group theory texts. An example of one of the many interesting topics covered by Davvaz is the group of arithmetic functions under Dirichlet convolution. This group is vitally important in number theory and is rarely discussed in any sort of depth in introductory algebra texts. Davvaz devotes an entire chapter to the topic.

 

One of the most frustrating aspects of this book is its unevenness. At one extreme, the book's discussion of modular arithmetic is in terms of clock arithmetic and literally features a picture of two clocks illustrating that five o'clock plus ten hours is three o'clock. The book has similarly elementary discussions of bijections and matrix multiplication. At the other extreme, many sections of the book are written in a way that assumes the reader is already familiar with field theory and vector spaces over fields. The chapter on matrix groups gives the formula for the order of the general linear group over a finite field, for example, yet the field axioms are never stated in the text. Another section begins "Let V be a finite-dimensional vector space over a field F," which seems out of place in a textbook with a preface that claims it is suitable for students having no prior exposure to algebra.

 

Unfortunately, this book contains many typographical mistakes, grammatical errors, and instances of undefined notation. Even worse, many of these mistakes affect the reader's ability to make sense of the text. For example, the definition of a translation, given on page 34, states that "A translation is an object from one location to another, without any change in size or orientation." Similarly, on page 111 Davvaz introduces an equivalence relation on \(S_n\) by saying that \(x\equiv_\sigma y\) if and only if there is an integer \(k\) such that \(x=x\sigma^k\). This is a minor issue of course, but it is clear that the book would have been greatly improved with better copy-editing.

 

Although there is a lot to like about Davvaz's A first course in group theory, I find it hard to believe that many instructors will adopt the book for their classes. For those interested in a matrix-heavy introduction to group theory, I would recommend looking at Armstrong's Groups and Symmetry.

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Benjamin Linowitz (benjamin.linowitz@oberlin.edu) is an Assistant Professor of Mathematics at Oberlin College. His website can be found at http://www2.oberlin.edu/faculty/blinowit/.