Preface

This book is intended for a one semester introduction to abstract algebra. Most introductory textbooks on abstract algebra are written with a two semester course in mind. See, for example, the books listed in the Bibliography below. These books are listed in approximate order of increasing difficulty. A search of the library using the keywords abstract algebra or modern algebra will produce a much longer list of such books. Some will be readable by the beginner, some will be quite advanced and will be difficult to understand without extensive background. A search on the keywords group and ring will also produce a number of more specialized books on the subject matter of this course. If you wish to see what is going on at the frontier of the subject, you might take a look at some recent issues of the journals Journal of Algebra or Communications in Algebra which you will find in our library.

Instead of spending a lot of time going over background material, we go directly into the primary subject matter. We discuss proof methods and necessary background as the need arises. Nevertheless, you should at least skim the appendices where some of this material can be found so that you will know where to look if you need some fact or technique.

Since we only have one semester, we do not have time to discuss any of the many applications of abstract algebra. Students who are curious about applications will find some mentioned in Fraleigh and Gallian . Many more applications are discussed in Birkhoff and Bartee and in Dornhoff and Horn .

Although abstract algebra has many applications in engineering, computer science and physics, the thought processes one learns in this course may be more valuable than specific subject matter. In this course, one learns, perhaps for the first time, how mathematics is organized in a rigorous manner. This approach, the axiomatic method, emphasizes examples, definitions, theorems and proofs. A great deal of importance is placed on understanding. Every detail should be understood. Students should not expect to obtain this understanding without considerable effort. My advice is to learn each definition as soon as it is covered in class (if not earlier) and to make a real effort to solve each problem in the book before the solution is presented in class. Many problems require the construction of a proof. Even if you are not able to find a particular proof, the effort spent trying to do so will help to increase your understanding of the proof when you see it. With sufficient effort, your ability to successfully prove statements on your own will increase.

We assume that students have some familiarity with basic set theory, linear algebra and calculus. But very little of this nature will be needed. To a great extent, the course is self-contained, except for the requirement of a certain amount of mathematical maturity. And, hopefully, the student’s level of mathematical maturity will increase as the course progresses.

I will often use the symbol \(\blacksquare\)  to indicate the end of a proof. Or, in some cases, \(\blacksquare\) will indicate the fact that no more proof will be given. In such cases the proof will either be assigned in the problems or a reference will be provided where the proof may be located. This symbol was first used for this purpose by the mathematician Paul Halmos.

Note: when teaching this course I usually present in class lots of hints and/or outlines of solutions for the less routine problems.

This version includes a number of improvements and additions suggested by my colleague Milé Krajčevski.