1.1. From the Preface.

The idea to write this book, and more important the desire to do so, is a direct outgrowth of a course I gave in the academic year 1959-1960 at Cornell University. The class taking this course consisted, in large part, of the most gifted sophomores in mathematics at Cornell. It was my desire to experiment by presenting to them material a little beyond that which is usually taught in algebra at the junior-senior level.

I have aimed this book to be, both in content and degree of sophistication, about halfway between two great classics, A Survey of Modern Algebra by Birkhoff and Mac Lane and Modern Algebra by Van der Waerden.

The last few years have seen marked changes in the instruction given in mathematics at the American universities. This change is most notable at the upper undergraduate and beginning graduate levels. Topics that a few years ago were considered proper subject matter for semi-advanced graduate courses in algebra have filtered down to, and are being taught in, the very first course in abstract algebra. Convinced that this filtration will continue and will become intensified in the next few years, I have put into this book, which is designed to be used as the student's first introduction to algebra, material which hitherto has been considered a little advanced for that stage of the game.

There is always a great danger when treating abstract ideas to introduce them too suddenly and without a sufficient base of examples to render them credible or natural. In order to try to mitigate this, I have tried to motivate the concepts beforehand and to illustrate them in concrete situations. One of the most telling proofs of the worth of an abstract concept is what it, and the results about it, tells us in familiar situations. In almost every chapter an attempt is made to bring out the significance of the general results by applying them to particular problems. For instance, in the chapter on rings, the two-square theorem of Fermat is exhibited as a direct consequence of the theory developed for Euclidean rings.

The subject matter chosen for discussion has been picked not only because it has become standard to present it at this level or because it is important in the whole general development but also with an eye to this "concreteness." For this reason I chose to omit the Jordan-Hölder theorem, which certainly could have easily been included in the results derived about groups. However, to appreciate this result for its own sake requires a great deal of hindsight and to see it used effectively would require too great a digression. True, one could develop the whole theory of dimension of a vector space as one of its corollaries, but, for the first time around, this seems like a much too fancy and unnatural approach to something so basic and down-to-earth. Likewise, there is no mention of tensor products or related constructions. There is so much time and opportunity to become abstract; why rush it at the beginning?

A word about the problems. There are a great number of them. It would be an extraordinary student indeed who could solve them all. Some are present merely to complete proofs in the text material, others to illustrate and to give practice in the results obtained. Many are introduced not so much to be solved as to be tackled. The value of a problem is not so much in coming up with the answer as in the ideas and attempted ideas it forces on the would-be solver. Others are included in anticipation of material to be developed later, the hope and rationale for this being both to lay the groundwork for the subsequent theory and also to make more natural ideas, definitions, and arguments as they are introduced. Several problems appear more than once. Problems, which for some reason or other seem difficult to me, are often starred (sometimes with two stars). However, even here there will be no agreement among mathematicians; many will feel that some unstarred problems should be starred and vice-versa.

Naturally, I am indebted to many people for suggestions, comments and criticisms. To mention just a few of these: Charles Curtis, Marshall Hall, Nathan Jacobson, Arthur Mattuck, and Maxwell Rosenlicht. I owe a great deal to Daniel Gorenstein and Irving Kaplansky for the numerous conversations we have had about the book, its material and its approach. Above all, I thank George Seligman for the many incisive suggestions and remarks that he has made about the presentation both as to its style and to its content. I am also grateful to Francis McNary of the staff of Ginn and Company for his help and cooperation. Finally, I should like to express my thanks to the John Simon Guggenheim Memorial Foundation; this book was in part written with their support while the author was in Rome as a Guggenheim Fellow.

1.2. Review by: Daniel Zelinsky.
Mathematical Reviews MR0171801 (30 #2028).

This text covers most of the basic material on abstract algebra, with a more or less classical approach. The exposition is pitched about at the level of Birkhoff and Mac Lane, though the mathematical content is more extensive and slightly more sophisticated. The book starts with set theory, a smattering of number theory, and then plunges into a short course on finite groups, ending with the existence of Sylow groups. A few commutative diagrams are visible. Next come rings and ideals, fields of quotients, polynomial rings, vector spaces and modules, including dual spaces, inner product spaces and the fundamental theorem on finitely generated modules over Euclidean rings (without uniqueness statements). Then the elementary theory of fields (surprise: a proof of the transcendence of e), ruler and compass constructions, Galois theory and solvability by radicals.

Almost the last third of the book is devoted to linear transformations and quadratic forms. The Jordan canonical form is done by reduction to nilpotent transformations and direct decomposition of the vector space; this is followed by the rational canonical form, done independently as a consequence of the fundamental theorem on modules over a polynomial ring. Then come traces, transposes (of matrices, dual spaces do not appear here), determinants; Hermitian, unitary and normal transformations; real quadratic forms.

Finally, there is a short chapter on three topics that have led to extensive mathematical research: Wedderburn's theorem on finite division rings, the classification of all (i.e., all three) real division algebras, and the four squares theorem (beginning of Waring's problem).

One gets the impression that in writing this text, the author was not interested in the content of the theorems alone, but also in the mathematical experience to which the student would be exposed, thus explaining several deliberate instances of inefficient proofs (the most significant was mentioned above: the independent reductions to Jordan and rational canonical form, without deriving either as a corollary of the other).

1.3. Review by: Jens Erik Fenstad.
Nordisk Matematisk Tidskrift 13 (4) (1965), 158-159.

This book, "aimed ... both in content and degree of sophistication, about halfway between two great classics, A survey of modern algebra by Birkhoff and Mac Lane and Modern algebra by Van der Waerden", is intended as a first introduction to algebra at university level. The choice of subject is relatively traditional, but the presentation is very thorough and the material is very well illustrated through examples. It offers a rich material for assignments.

After an introductory chapter on set theory, mappings and the integers, there follows a long and very detailed section on groups, where the author particularly seeks to highlight proof methods characteristic of this theory (finite groups). This is followed by a chapter on rings. The theory of ideals is developed only to a modest extent, while the theory of Euclidean rings is presented more extensively (including a proof, via the theory of Gaussian integers, of the well-known Fermat's theorem that every prime number of the form $4n+1$ can be written as a sum of two squares).

Before the chapter on field theory comes a section on vector spaces and modules (including the basis theorem for finite abelian groups). Galois theory, following Artin's example, is given a clear presentation, and the usual applications to the theory of equations are included. There is also a larger section on linear transformations, and a concluding chapter deals with some selected topics (Wedderburn's theorem on finite division rings, Frobenius' characterisation of division rings algebraic over the real numbers, and Lagrange's theorem that every positive integer can be written as a sum of four squares).

The book is lively and well-written, and there are plenty of non-trivial results in terms of definitions and terminology. It would certainly be excellent as a first book in algebra.

1.4. Review by: Robert Ellis.
The American Mathematical Monthly 74 (6) (1967), 759.

This is an excellent text designed for an honours section in algebra, but the manner in which it is written makes it possible to teach a course at several different levels. By amplifying the proofs, going over them thoroughly in class, and assigning the problems intended to illustrate the theory, one can gear the course to the average student. The above average student could be encouraged to work on the "not too obvious problems." Another good exercise is to have the students puzzle out the proofs for themselves.

The book has many attractive features. The problems are excellent, varying from the immediate to those one had best not assign unless he had a few weeks spare time to work on them himself. Many theorems are proved in two or three different ways and in varying degrees of generality depending upon the context in which they occur. Most important, the author demands a lot of the student; and, if the instructor supports the author in this demand, the student will learn a great deal from the book.

Most important of all, I enjoyed teaching a course with this text.

2.1. From the Preface.

This book is not intended as a treatise on ring theory. Instead, the intent here is to present a certain cross-section of ideas, techniques and results that will give the reader some inkling of what is going on and what has gone on in that part of algebra which concerns itself with noncommutative rings. There are many portions of great importance in the theory which are not touched upon or which are merely mentioned in passing. On the other hand there is a rather detailed treatment given to some aspects of the subject.

While the account given here is not completely self-contained, to follow it does not require a great deal beyond a good first course in algebra. Perhaps I should spell out what I would expect in such a course. To begin with one should have been introduced to some of the basic structures of algebra-groups, rings, fields, vector spaces-and to have seen some of the basic theorems about them. One would want a good familiarity with homomorphisms, the early homomorphism theorems, quotient structures and the like. One should have learned with some thoroughness linear algebra - the fundamental theorems about linear transformations on a vector space. This type of material can be found in many books, for instance, Birkhoff and Mac Lane A Survey of Modern Algebra or my book Topics in Algebra.

Beyond these standard topics cited above I shall make frequent use of results from the theory of fields. All these can be found in the chapter on field theory in van der Waerden's Modern Algebra. My advice, to the reader not familiar with this material, is to read into a proof until such a result is cited and then to read about the notions arising in van der Waerden's book. Finally, I shall continually use Zorn's Lemma and the axiom of choice.

A great deal of what is done in this book is based on selected parts of two sets of my notes published in the University of Chicago lecture notes series. Part of this selection and weeding process, polishing and blending together was accomplished in a course I gave at Bowdoin College, under the auspices of the Mathematical Association of America, in the summer of 1965 to a group of mathematicians teaching at various colleges and smaller universities. I should like to thank the participants in that course for their patience and enthusiasm. There are many others I should like to thank, Nathan Jacobson and Irving Kaplansky, for the part they and their work have played in my formation as a mathematician, Shimshon Amitsur for the many pleas-ant hours spent together working and discussing ring theory and my students, Claudio Procesi and Lance Small, for taking the notes at Bowdoin and for their stimulating comments, suggestions and improvements.

2.2. From the Contents.

1. THE JACOBSON RADICAL.
1. Modules
2. The radical of a ring
3. Artinian rings
4. Semisimple Artinian rings
References

2. SEMISIMPLE RINGS
1. The density theorem
2. Semisimple rings
3. Applications of Wedderburn's theorem
References

3. COMMUTATIVITY THEOREMS
1. Wedderburn's Theorem and some generalizations
2. Some special rings
References

4. SIMPLE ALGEBRAS
1. The Brauer group
2. Maximal subfields
3. Some classic theorems
4. Crossed products
References

5. REPRESENTATIONS OF FINITE GROUPS
1. The elements of the theory
2. A theorem of Hurwitz
3. Applications to group theory
References

6. POLYNOMIAL IDENTITIES
1. A result on radicals
2. Standard identities
3. A theorem of Kaplansky
4. The Kurosh Problem for P.I. algebras
References

7. GOLDIE'S THEOREM
1. Ore's theorem
2. Goldie's theorems
3. Ultra-products and a theorem of Posner
References

8. THE GOLOD-SHAFAREVITCH THEOREM
References

2.3. Review by: N Divinsky.
The American Mathematical Monthly 75 (7) (1968), 804-805.

This seems to be an excellent text for a first year graduate course on ring theory. It should follow a sound introductory course in abstract algebra and if such a prerequisite is available at the junior year then this can be used for undergraduate seniors.

It begins with the Jacobson radical, done via modules, obtains the Wedderburn results for rings with descending chain conditions, the Jacobson results on general rings, the Goldie theorems for rings with ascending chain conditions, and closes with the Golod-Shafarevich theorem which separates locally nilpotent algebras from nilalgebras and gives a negative answer to the general Burnside problem. In between there are pithy sections on rings with polynomial identities, commutativity theorems (Herstein's personal specialty), representations of finite groups and division algebras.

The book is up to date, easy to read, well motivated and filled with fine insights and appealing open questions.

2.4. Review by: Wallace S Martindale III.
Mathematical Reviews MR0227205 (37 #2790).

This colourful and informative book on noncommutative ring theory is based on a series of expository lectures given by the author in the summer of 1965 at Bowdoin College before an audience of teachers from colleges and small universities. These lectures were in turn based to a large extent on the author's 1961 and 1965 University of Chicago notes on ring theory. The author's statement in the preface that a good first course in algebra (plus some field theory for Chapter 4) is a prerequisite for the reading of this book is a reasonable one. The level at which the book is pitched falls in this reviewer's opinion somewhere between N H McCoy's The theory of rings (1964) and J Lambek's compactly written Lectures on rings and modules (1966).

In the course of the first two chapters the Jacobson structure theory for arbitrary rings is developed at a leisurely pace, with the Wedderburn-Artin theory of finite-dimensional algebras and rings with minimum condition falling out as a by-product. Already, however, the reader will have become aware of why this is an exciting book - the author's ability to smoothly shift back and forth between the flow of the general theory, on the one hand, and non-trivial applications, striking examples, recent results, and open questions on the other. At times a major application of the theory is treated in considerable depth, as we see when the Wedderburn theorems are used in giving a complete proof of Burnside's theorem that a torsion group of matrices over a field is locally finite. At other times we are quickly rushed to the frontier of the subject and then back again - an instance of this occurs when we are presented with the author's own recent elementary counterexample to the conjecture that the intersection of the powers of the Jacobson radical of a right Noetherian ring is zero.

The remaining chapters of the book are all independent of each other, except that part of Chapter 7 depends on Chapter 6. Chapter 3 is in a sense the most purely ring theoretic one: the full force of the Jacobson structure theory is brought to bear on developing several commutativity theorems. ...
...
The spirit of the Carus Monograph series is clearly embodied in this moving and excellently written account of important aspects of classical and modern ring theory. The book will undoubtedly be a popular one for a wide class of mathematicians and students.

3.1. From the Preface.

These notes are based on a course I gave at the University of Chicago in the Spring Quarter of 1964. They were originally published in 1965 in the Lecture Notes of the Mathematics Department of the University of Chicago under the title of "Topics in Ring Theory." This present version is a reworking of these notes - many of the proofs in the first chapters have been changed and some material has been added.

The subject matter divides itself naturally into three distinct parts. In the first three chapters I treat the structure of a simple associative ring and some of its subsets as Lie and Jordan rings and use these results to obtain theorems about the associative nature of the ring itself. Chapters 4, 5, 6 are devoted to the theorems of Goldie and some of their consequences for rings conditioned by some appropriate chain condition. The material of the last chapter is concerned with recent examples and counter-examples; we find there the Golod-Shafarevitch theorem and examples due to Bergman and Sasiada.

The overlap of the material here and in my Carus Monograph "Non-Commutative Rings" is not large; it consists of one of the proofs of Goldie's theorem, ultra-products and Posner's theorem and the Golod-Shafarevitch theorem.

3.2. Review by: Wallace S Martindale III.
Mathematical Reviews MR0271135 (42 #6018).

These notes are essentially the publication in (paperback) book form of the author's 1965 University of Chicago Mathematics Lecture Notes publication of the same title. The book is an account of three distinct topics in ring theory.

The first three chapters are an exposition of an area originated and developed by the author in the early 1950's, namely, the Lie and Jordan structure of simple associative rings (without finiteness conditions, often with involution). Part of the motivation for this study (carried out in Chapters 1 and 2) is to show that simplicity of an associative ring with involution is all that is needed to prove that the related Lie ring of skew elements (and its derived ring) are essentially also simple, thereby shedding new light on some results about classical Lie algebras. The third chapter leads up to the result that Jordan homomorphisms onto prime rings must be either homomorphisms or anti-homomorphisms. The main point we wish to emphasise, however, is that in these first three chapters the reader witnesses clearly the distinctive style which characterises much of the author's approach to ring theory - that of obtaining interesting results in general settings by means of a series of elementary lemmas rather than by recourse to finite assumptions and highly developed structure theory.

Chapters 4, 5, and 6 deal with non-commutative Noetherian rings, including a couple of accounts of Goldie's theorems, results about nil subrings of Noetherian rings, Posner's theorem, and the Lesieur-Croisot tertiary decomposition theory. The last chapter discusses important counterexamples due to Golod and Shafarevich, Sasiada, and Bergman. A good portion of Chapters 4, 5, and 7 are also to be found in the author's recent book Noncommutative rings (1968).

4.1. From the Preface.

In his poem "A Grammarian's Funeral," Robert Browning, in describing the ills and woes that befell the old scholar, writes "Calculus racked him." True, the calculus to which Browning refers is kidney stones, but it could very well be the differential and integral calculus as it is too often presented to the poor innocent learning it for the first time.

There has been a tendency in the last 10-15 years to do the calculus for the beginner in a completely formal, rigorous, and to our minds-stilted way. This preoccupation with detail and technique of proof, rather than with the basic ideas involved, has made a dark mystery out of concepts which are essentially clear and simple. The student, unaware of the basic direction of the subject, confuses the detail of rigorizing an argument with the fundamental ideas being developed. In consequence, the average student, upon finishing his first year of calculus, has neither a full and coherent picture of what the subject is about, nor a mature understanding of mathematical technique and rigour. His ability to use the reasoning and results of the calculus - and let's never forget that the calculus was devised to handle large classes of problems in and out of mathematics - unfortunately is too limited by this formalistic approach.

Our motivation for writing this book is a desire to fight this recent trend. In this we certainly are not the first - in recent years a fine attempt in this direction was made by Serge Lang. It should be pointed out that the calculus was taught unrigorously for the largest part of this century; unrigorously, but, unfortunately, too often sloppily. One can give an intuitive approach to the subject which is honest and correct without being sloppy, if one limits one's attention to well-behaved functions. Fortunately, such well-behaved functions exist in abundance and are, in fact, the functions that arise in the basic material of mathematics, engineering, physics, chemistry, and many other fields.

We have tried to be completely intuitive and to motivate the various ideas and concepts introduced. But there are times when one must get down to the fine points of an argument in order to give an honest presentation. This takes place, for instance, in proving the fundamental theorem of the integral calculus. We feel that most readers could easily skip that part without substantially hurting their understanding of the subject matter, since that section was preceded by an intuitive discussion of the integral.

There are other places where greater precision and rigour would have made the writing easier for us - albeit more difficult for the reader - especially in the parts dealing with infinite series. All in all, we have tried to keep a uniformly intuitive approach, paying as little heed to formalism and rigour as possible without distorting matters . We have included many exercises . It took a conscious effort on our part not to include many tricky, clever, or cute problems. Instead, we have put in problems that would illustrate the techniques discussed in as straightforward a manner as possible. Although we do have some exercises that come from outside of mathematics, by and large the problems are what one could describe as mathematical ones.

It is our belief that if the student assimilates and understands the basic mathematical ideas and techniques exposed, he will have no difficulty in adapting them to problems arising in his own discipline - where they will arise for him in a more natural and meaningful way than we could possibly provide if he is not a mathematics major. We have deliberately only treated the one - variable case . Firstly, the plausibility arguments and the intuitive reasoning used here can very often be made into a precise discussion; the multivariable case is much subtler and a good plausibility argument may be very far from the truth there even for the best behaved situations. At any rate, by the time the student arrives at the study of the multivariable calculus he is mature enough to be able to withstand the rigours of rigour and can be taught the subject with a great deal of precision. There are many excellent books giving a good treatment of the multivariable calculus.

One final word about the book. We have deliberately tried to keep it short. The psychological effect on a student setting out on what seems to him a rough voyage, of a huge, weighty tome of about 1000 pages, is devastating. In fact, it is on us too. We feel that we left out very little that is essential or that can't be picked up easily if it is needed.

4.2. Review by: James V Lewis.
The American Mathematical Monthly 80 (1) (1973), 90-91.

This book is intended to provide a one-year introduction to single variable calculus which is intuitive and correct. The authors use an informal readable style. There are excellent intuitive introductory discussions of the derivative as a rate of change, the use of limits in graphing, and upper and lower sums for the definite integral. The book is almost free of error. Unfortunately, despite these excellent features the book has serious weaknesses in its organisation.

After the very clear examples which would make it easy to introduce the integral as a sum, the authors lose courage and define the integral in terms of antiderivatives. If the student accepts this then the statement of the Fundamental Theorem, Part II (after a long rigorous proof) is pointless since it holds by definition. Students would more easily understand the definition of the integral as a sum (using area as the interpretation) followed by an intuitive proof of the Fundamental Theorem and its application to evaluating integrals by antiderivatives.

This book has numerous other deficiencies: too few worked examples; too few easy drill problems; too little use of numerical approximation to get the feel of concepts of limit, derivative and integral; almost no discussion of the derivative as the slope of a curve or of the shape of a curve from the signs of the first and second derivatives; lack of applications in the social sciences and the solution of the simplest differential equations (except by series); lack of a table of integrals (or practice in its use). The typography and layout actually conceal key formulas, results and summaries. After using this book in three classes in our two semester calculus sequence at the University of New Mexico, we recommend it as a source of interesting examples for class discussion, but not as a class text.

4.3. Review by: J S Fowlie.
The Mathematical Gazette 56 (396) (1972), 139-140.

This American text provides an intuitive approach to one-variable calculus in an attempt to escape from the excess of rigour which, the authors feel, has led to loss of understanding of the basic concepts and inability to apply them. The approach in this country, usually directed at less mature pupils, has generally not been guilty of this, and the treatment may still seem unduly rigorous. For example there is a lengthy, if informal, discussion of continuity and the existence of limits before much of the basic technique of calculus is introduced.

Lack of rigour is justified by restricting attention to "well-behaved" functions; it is not always clear whether the results obtained are valid only for such functions, or the methods of obtaining them. The reader's intuitive powers would be better developed if he were given more examples of functions which "misbehave".

The authors seem to feel that their approach demands an informality of language (why does O.K. sometimes appear as okay?). In their preface they state that "... greater precision and rigour would have made the writing easier ...". There is confusion between precision of language and precision of mathematical argument: sentences such as "this seems to be a good time to stop for a moment and put what we've been up to on a somewhat more precise basis" may suit the lecture room but are irritatingly woolly in print. The book is well produced but the text tends to run on-easy to read, but difficult to use for reference even with the help of the index and section summaries. The authors, who are professors of mathematics at Chicago and Illinois respectively, seem determined to be "with it", choosing photographs of themselves in roll-neck sweaters, one with a cigarette between his lips, the other wearing a string of beads, for the dust-jacket!

The ground covered is differential and integral calculus of a single variable, going as far as Taylor's series, but not dealing with differential equations. The notation is traditional and the basis is the definition of the derived function as

          $\large\lim_{h \rightarrow 0} \Large \frac{f(x+h) - f(x)}{h}$,

provided the limit exists. The product formula is used to obtain the derivative of $x^{n}$, rather than the binomial theorem, and also the derivative of $\sin x$, assuming its existence. The integral as the anti-derivative is considered at great length in a way perhaps more appropriate to a university analysis course.

While the book may well be a valuable contribution in the American context it has little relevance to the British scene.

5.1. From the Preface.

This book is based on notes prepared for a course at the University of Chicago. The course was intended for non-majors whose mathematical training was somewhat limited.

Our aim is to cover a selection of topics that give something of the flavour of modern mathematics. Furthermore, we try to carry each topic far enough to prove something substantial. Mastery of the material requires nothing beyond the algebra and geometry normally covered in high school.

In transforming the notes into a book, we expanded the material considerably. As a result, we feel that the book may be suitable for a wider audience. For instance, it could be used in courses designed for students who intend to teach mathematics. Several of the topics touch directly on matters that the prospective teacher might be teaching his own students: geometry, probability and game theory, and portions of the number theory. Another possible use of the book is in a course for mathematics majors as a precursor to more specialised courses. Permutations, groups, and infinite sets could serve as their introduction to abstract mathematics.

We want the reader to see mathematics as a living subject in which new results are constantly being obtained. Whenever possible, we mention the most recent advances in the area under discussion.

Number theory has fascinated mankind for centuries. The subject abounds in open questions that are easily stated and easily understood. We open our long Chapter 2 with a preliminary informal section that surveys some of these challenges. This should motivate the more careful treatment that follows.

In much of modern mathematics, one studies abstract systems. This is done not for the sake of generality alone, but because it has been found to be effective in solving classical problems. Abstractions evolved from special situations. A notable example is the transition from permutations to groups. Chapters 3 and 4 illustrate this: a "concrete" special case serves as a prelude to an abstraction. Permutations are easily grasped. They lend themselves to a variety of charming experiments; and nontrivial theorems and challenging open questions are not at all lacking. In this way, the ground is prepared for the general concept of a group.

In Chapters 5 and 6, we treat two topics that are nearly self-contained: finite geometries and games. This is very much twentieth-century mathematics, and yet a minimum of technique is needed as a prerequisite. The high spot of Chapter 5 is the Bruck-Ryser theorem; that of Chapter 6 is the theory of two-by-two games.

Many students are introduced to set theory in grade school or high school. Chapter 1 reviews this material for such students and establishes the language we use throughout the book. We felt that it would be interesting to push beyond this modest level into the Cantor theory of infinite sets. People find infinity mysterious. The little that Chapter 7 contains on the cardinal equivalence of infinite sets sheds some light on these mysteries and introduces the reader to a remarkable chapter of current mathematics

We try to avoid manipulation and technical details as much as possible, keeping the central ideas in the foreground. This cannot always be done. Getting to the crux of a substantial piece of mathematics requires hard work and technique. But we believe that the book is within the reach of people with the background we mentioned in the outset.

We owe a considerable debt to our colleague Melvin Rothenberg, who collaborated in the initial stages of the project, and to numerous students who made valuable comments and suggestions. In connection with the Josephus permutation (Chapter 3), we are grateful to Victor Akylas, Michael Bix, Alex Kaplansky, and Lucy Kaplansky for helpful hand computations, and to Raphael Finkel for preparing and running a computer program at the University of Chicago.

5.2. Review by: Kenneth C Skeen.
The Mathematics Teacher 68 (5) (1975), 409.

Here's a nicely done book for an appropriate level of study. Chapter topics include sets and functions, number theory, permutations, group theory, 'finite geometry, game theory, and infinite sets.

The authors suggest it for "non-majors whose mathematical training was somewhat limited." Later, after expanding the original course, they add "a wider audience," including prospective teachers.

Topics and methods of presentation seem much more appropriate for the latter audience, who, one might anticipate, would enter the course with two or three years of study in a mathematical major. For these, it would seem highly interesting.

6.1. From the Preface.

In addition to the proof previously given for the existence, two other proofs of existence are carried out. One could accuse me of overkill at this point, probably rightfully so. The fact of the matter is that Sylow's theorem is important, that each proof illustrates a different aspect of group theory and, above all, that I love Sylow's theorem. The proof of the conjugacy and number of Sylow subgroups exploits double cosets. A by-product of this development is that a means is given for finding Sylow subgroups in a large set of symmetric groups.

For some mysterious reason known only to myself, I had omitted direct products in the first edition. Why is beyond me. The material is easy, straightforward, and important. This lacuna is now filled in the section treating direct products. With this in hand, I go on in the next section to prove the decomposition of a finite abelian group as a direct product of cyclic groups and also prove the uniqueness of the invariants associated with this decomposition. In fact, this decomposition was already in the first edition, at the end of the chapter on vector spaces, as a consequence of the structure of finitely generated modules over Euclidean rings. However, the case of a finite group is of great importance by itself; the section on finite abelian groups underlines this importance. Its presence in the chapter on groups, an early chapter, makes it more likely that it will be taught.

Another entire section has been added at the end of the chapter on field theory. I felt that the student should see an explicit polynomial over an explicit field whose Galois group was the symmetric group of degree 5, hence one whose roots could not be expressed by radicals. In order to do so, a theorem is first proved, which gives a criterion that an irreducible polynomial of degree $p$, $p$ a prime, over the rational field has $S$, as its Galois group. As an application of this criterion, an irreducible polynomial of degree 5 is given, over the rational field, whose Galois group is the symmetric group of degree 5.

There are several other additions. More than 150 new problems are to be found here. They are of varying degrees of difficulty. Many are routine and computational, many are very difficult. Furthermore, some interpolatory remarks are made about problems that have given readers a great deal of difficulty. Some paragraphs have been inserted, others rewritten, in places where the writing had previously been obscure or too terse.

Above I have described what I have added. What gave me greater difficulty about the revision was, perhaps, that which I have not added. I debated for a long time with myself whether or not to add a chapter on category theory and some elementary functions, whether or not to enlarge the material on modules substantially. After a great deal of thought and soul-searching, I decided not to do so. The book, as it stands, has a certain concreteness about it with which this new material would not blend. It could be made to blend, but this would require a complete reworking of the material of the book and a complete change in its philosophy - something I did not want to do. A more recent addition of this new material, as an adjunct with no applications and no discernible goals, would have violated my guiding principle that all matters discussed should lead to some clearly defined objectives, some highlighting, and some exciting theories. Thus, I decided to omit the additional topics.

Many people wrote to me about the first edition, pointing out typographical mistakes or making suggestions on how to improve the book. I should like to take this opportunity to thank them for their help and kindness.

The most substantial of the many revisions are in the chapter on group theory.

7.1. From the Preface.

I have tried to give in this book a rather intense sampler of the work that has been done recently in the area of ​​rings endowed with an involution. There has been a lot of work done on such rings lately, in a variety of directions. I have not attempted to give the last-minute results, but, instead, I have attempted to present those whose statements and proofs typify the kind of things that are being done.

The results from this area have important applications outside of associative ringing theory itself. One such broad area of ​​application is to the theory of Jordan algebras and of quadratic Jordan algebras. I have not developed this interrelationship here, not because it isn't interesting or relevant but because it would have taken me too far afield. It also would have expanded the book immensely. We recommend that the reader interested in this connection go to the mathematical literature, especially to the papers by N Jacobson, K McCrimmon, and M Osborn, where such applications are readily found.

A second area of ​​applications is that of operator algebras and Banach algebras. Here, too, for the reasons cited above, I decided not to develop these applications here. The interested reader will find such connections in the papers by Miers, Sunouchi, and Topping and in the Springer Lecture Notes by de la Harpe.

7.2. Review by: Yurii N Maltsev.
Mathematical Reviews MR0442017 (56 #406).

In this book there is given a comprehensive account on the main results of the theory of associative rings with involution obtained recently. It consists of six chapters: Ring-theoretic preliminaries, Regularity conditions on skew and symmetric elements, Commutativity theorems, Mapping theorem, Polynomial identities, Potpourri.

8.1. Review by: Johnston Anderson.
The Mathematical Gazette 64 (427) (1980), 61-62.

Let me begin by praising this book. The authors are distinguished, indeed, in at least one case, quite famous; the printing is excellent and the misprints and errors seem to be few; the diagrams are clear and the binding is sound.

Unfortunately, these virtues hardly compensate for the contents. I have now read this book three times. After the first perusal, I found myself so incensed that I put the book aside and did not even attempt a review, feeling that I should wait until my reactions had cooled and hoping that maybe the book would improve with age, like port. Instead, it rather grows on one with the passage of time, like mildew.

To quote the authors, the original target audience were "non-specialists whose mathematical training is somewhat limited" but, with the growth of the text from lecture notes to book, the authors feel confident to include within its ambit would-be teachers of mathematics and even those who will go on to specialise in mathematics. The contents, also according to the preface, "aim to cover a selection of the topics that give something of the flavour of modern mathematics" (my italics).

The chapter headings (in order) are Sets and functions (that well known keystone of modern mathematics), Number theory (?), Permutations, Group theory, Finite geometry, Game theory and Infinite sets. It is thus immediately apparent that there is the usual arrogant presumption that "modem mathematics" really means abstract pure mathematics or, even worse, as seems to be the case here for these well known algebraists, that the only modern mathematics worth the name is that which is algebra-related. This is a poisonous and pernicious doctrine which enjoys all-too-widespread support in the UK, too. It is enshrined in A level syllabuses. I have long wished to see expunged from A level examinations such options as "Algebraic structure", with its artificial and pointless questions; the underlying ideas may have a worthwhile place in the curriculum, as a non-examinable topic which enlightens other parts of mathematics and reveals patterns. [I know the old argument that if it isn't examinable it won't be covered; my answer is "Too bad: I'd rather have it not covered than examined".] I speak as a pure mathematician, but one who is aware of the great British tradition of producing applied mathematicians of quality. To me, "modem mathematics" should include graph theory and combinatorics, statistics (and probability), algorithms (and perhaps computing), mathematical modelling and even perhaps such things as information theory. That is quite apart from the question of what sort of mathematics the target audiences mentioned earlier should have.

My disillusionment, frustration and irritation is not all philosophical, however; it extends to the detailed execution of the text as well. There is a spurious attempt to make the poor, benighted non-specialist feel at home: little biographical snippets printed in a different fount and a different size, just sufficient to divert the reader from his train of thought, smatterings of history that are not really historical in the proper sense but catalogues of dates and events (who on earth wants to know this without being aware of the historical mathematical background?), efforts at breathless immediacy ("in 1973, Hagis and McDaniel showed that an odd perfect number must be larger than $10^{50}$ and have a factor of at least 11213 "), examples on sets (American presidents, baseball pitchers, readers of Harper's and Playboy), which are not only twee, but culture-dependent, and also the kind of thing that students find both unrealistic and patronising. Yet despite these efforts the underlying format is still the same boring old Definition-Theorem-Proof that we know so well and which even specialist mathematicians find difficult. Examples usually illustrate rather than motivate-in a book so leaning towards abstraction, one would have thought that motivation through examples for concepts and definitions might have been more useful. Furthermore, the treatment of both formal and informal parts of the text is uneven; sometimes, laborious detail is used to demonstrate some fairly elementary point while, at other times, the terse, concept-dense sentences of a typical textbook proof will be deployed without compunction.

If one may borrow the wonderfully pungent language of J M Hammersley, writing six years before the first edition of this book, this is yet another episode in the enfeeblement of traditional mathematical skills by modern mathematics and similar soft intellectual trash, geared to the imagined needs of liberal-arts-type students.

8.2. Review by: Benno Artmann.
Mathematical Reviews MR0497399 (58 #15763).

The book originates from lectures for non-majors in mathematics which were expanded to give a text which could also be used as general introductory material for mathematics students, especially for future teachers. It contains chapters on: (1) Sets and functions, (2) Number theory, (3) Permutations, (4) Groups, (5) Finite geometry, (6) Game theory, and (7) Infinite sets. Before going into detail, the reviewer wishes to stress that he thinks that the authors have succeeded very well in creating an adequate picture of mathematics for their audience, and that the topics chosen are optimal for their intentions. Chapter 1 gives a review for those students who do not know the elementary notions concerning sets and functions. Maybe the addition of some diagrams would be helpful for the beginner. {The reviewer wonders whether one should not use the standard terms injective, surjective, bijective. A minor, but useful, change would be to use $id$ instead of $i$ for the identity function and to treat this symbol like exp, sin. The equivalence of bijectivity and the existence of an inverse function would make a nice theorem; it deserves a better place than an exercise.} On the whole, this chapter could be shorter than 30 pages.

Chapter 2 is a very lively introduction to number theory, culminating in sections on Waring's problem and Fermat primes. After an informal introduction on prime numbers, the concept of a ring is defined, $\mathbb{Z}$is characterised and then the theory of divisibility for integers is developed. The blend of numerical examples and theorems is excellent. {The reviewer has some reservations about the explanation of (−1)(−1) = 1 (or, minus × minus = plus) in Section Three. The authors quote Stendhal who describes his difficulties at school in understanding this formula, and then proceed to prove the formula from the ring axioms. Surely this can be done, but it seems to miss the point. At school the problem is to define multiplication for negative integers in order to make $\mathbb{Z}$ a ring, and not deduction from the ring axioms. The confusion may become even greater by this explanation.}

Chapters 3 and 4 should be considered together. Even and odd permutations are explained, and as a very nice example the Josephus permutation is studied in detail. The treatment of group theory in Chapter 4 is somewhat less original than the other chapters. The standard beginning material up to Lagrange's and Cayley's theorems is presented, but the reviewer does not see what really matters here. It would be nice to have, say, some demonstration of the power of the group concept as the formalisation of "symmetry" or something like that.

In contrast, Chapter 5 on finite affine planes gives a substantial piece of that theory, up to the proof of the Bruck-Ryser theorem on the nonexistence of planes for some orders $n$. ...
...
8.3. Review by: Stuart J Sidney.
American Scientist 67 (2) (1979), 247.

Based on a course given at the University of Chicago, this is a cross between a text and an appreciation. As a text it offers a systematic exposition for the non-specialist of the elementary parts of several "noncontinuous" areas in the mainstream of modern mathematics, exercises included. As an appreciation, it offers "snapshots" of some of the deeper and more striking applications of the general theory (some of which may be unfamiliar to the average professional), a good deal of mathematical history, and a fine sense of mathematics as a continuously developing body of knowledge.

The chapter headings are: sets and functions, number theory, permutations, group theory, finite geometry, game theory, infinite sets. Since many other concepts appear in passing-for example, rings and probability-there is material on most key ideas which involve neither linear algebra nor topology (and limits). This is a good place for a general audience to get an idea of what mathematics is about today.

9.1. From the Preface.

In the last half-century or so, abstract algebra has become increasingly important not only in mathematics itself, but also in a variety of other disciplines. For instance, the importance of the results and concepts of abstract algebra plays an ever more important role in physics, chemistry, and computer science, to cite a few such outside fields.

In mathematics itself, abstract algebra plays a dual role: that of a unifying link between disparate parts of mathematics and that of a research subject with a highly active life of its own. It has been a fertile and rewarding research area both in the last 100 years and at the present moment. Some of the great accomplishments of our twentieth-century mathematics have been precisely in this area. Exciting results have been proved in group theory, commutative and noncommutative ring theory, Lie algebras, Jordan algebras, combinatorics, and a host of other parts of what is known as abstract algebra. A subject that was once regarded as esoteric has become considered fairly down-to-earth for a large cross-section of scholars

The purpose of this book is twofold. For those readers who either want to go on to do research in mathematics or in some allied fields that use algebraic notions and methods, this book should serve as an introduction - and, we stress, only as an introduction to this fascinating subject. For interested readers who want to learn what is going on in an engaging part of modern mathematics, this book could serve that purpose, as well as provide them with some highly usable tools to apply in the areas in which they are interested.

The choice of subject matter has been made with the objective of introducing readers to some of the fundamental algebraic systems that are both interesting and of wide use. Moreover, in each of these systems the aim has been to arrive at some significant results. There is little purpose served in studying some abstract object without seeing some nontrivial consequences of the study. We hope that we have achieved the goal of presenting interesting, applicable, and significant results in each of the systems we have chosen to discuss.

As the reader will soon see, there are many exercises in the book. They are often divided into three categories: easier, middle-level, and harder (with an occasional very hard one). The purpose of these problems is to allow students to test their assimilation of the material, to challenge their mathematical ingenuity, to prepare the ground for material that is yet to come, and to be a means of developing mathematical insight, intuition, and techniques. Readers should not become discouraged if they do not manage to solve all the problems. The intent of many of the problems is that they will be tried - even if not solved - for the pleasure (and frustration) of the reader. Some of the problems appear several times in the book. Trying to do the problems is undoubtedly the best way of going about learning the subject.

We have strived to present the material in the language and tone of a classroom lecture. Thus, the presentation is somewhat chatty; we hope that this will put the readers at ease. An attempt is made to give many revealing examples of the various concepts discussed. Some of these examples are carried forward to be examples of other phenomena that come up. They are often referred to as the discussion progresses.

We feel that the book is self-contained, except in one section - the second-last one of the book - where we make implicit use of the fact that a polynomial over the complex field has complex roots (that is the celebrated Fundamental Theorem of Algebra due to Gauss), and in the last section where we make use of a little of the calculus.

9.2. Review by: Georgia Benkart.
The American Mathematical Monthly 94 (8) (1987), 804-806.

Ever since it was first published 23 years ago, I N Herstein's Topics in Algebra has set a standard for texts on abstract algebra. A whole generation of textbooks and an entire generation of mathematicians, myself included, have been profoundly influenced by that text. (I never would have dreamt in my undergraduate days as I struggled in frustration with some of the book's exercises that years later, as a practicing algebraist, I would be asked to review its descendant, Abstract Algebra.) Since 1964, there have been a host of texts written on the subject of abstract algebra. Too many of those books have diluted the subject to the point that students arrive for graduate work ill-prepared to tackle the rigors of a graduate algebra course. Another recent trend has been to include some of abstract algebra's beautiful applications to codes, lattices, geometries, and crystallographic groups to help instructors and students answer the proverbial question, "What's all this theory good for?" and to meet the growing demand for training in applied algebra. The book under review, Herstein's Abstract Algebra, follows neither of these trends. It is abstract algebra - rigorous and pure.

Abstract Algebra is neither a remodelling nor a revision of the classic, Topics in Algebra. Rather, it is a new text on groups, rings, and fields intended to be less inclusive and more informal than Topics. The tone is more chatty, paralleling a classroom discussion in its presentation. Clear, concise exposition, which has been the hallmark of Herstein's texts, is at the heart of this new one. The student is sure to sense that behind all the theory presented is a mathematician trying above all else to communicate his subject and, in the process, to convey his great enthusiasm for it. Years of experience teaching algebra courses undoubtedly have made the author very sensitive to the difficulties that the student is likely to encounter. Care is taken to head mistakes off at the pass by presenting non-examples and discussing common misconceptions. The reader receives stern warnings such as the exhortation, "Avoid a mathematical stomachache later by assimilating this section well," but frequently sympathy is extended, too

Nowadays, the umbrella of modern algebra covers a wide diversity in both course content and level. To achieve greater flexibility, Abstract Algebra is designed for courses at three different levels of difficulty, and an outline of what might be included in each course is provided in the Instructor's Manual. To accommodate this scheme further, problems are divided into three categories - easier, middle-level, and harder - except for an occasional section where the exercises are intentionally not rated in order to give students experience in working problems of an unknown persuasion. Students having had the middle-level course done rigorously or the harder-level course, which is aimed a shade below an honours course, should be ready to tackle graduate algebra, especially if they have worked a goodly number of the exercises. Some graduate programs, however, expect students to be familiar with Galois theory, and as none is included in this text, a student may have to do a little catch-up reading from, say, Topics in Algebra.

Abstract Algebra is structured around a spiral approach to new concepts. Frequently, a concept is first introduced in an exercise in which the reader is asked to show that a simple property holds. Later, a formal definition is given, and a few results about it are developed. In subsequent sections where the concept is used heavily, the relevant facts are generally recalled before they are needed. With a few exceptions, some of which are discussed below, this is accomplished very effectively. Students are urged to go back and tackle old exercises with the new theories they have just learned. The structure of the text is such that this is a very worthwhile endeavour, and in fact, instructors could well encourage even more of this activity.

A course in algebra is often a student's first encounter with abstract mathematics, and a question that frequently arises from amidst all the definitions, lemmas, theorems, and proofs is "What am I expected to know?" This may be a point of some confusion with Abstract Algebra, where there are definitions and there are Definitions, and usually both are very important For example, image and index are left as informal definitions while kernel is afforded a formal Definition. The left coset is discussed in much detail, but the right coset is relegated to the exercises. Without warning, ten pages later in a section on normal subgroups, the reader is assumed to be familiar with both kinds of cosets. A brief notice after the exposition on left cosets directing the reader to the exercises on right cosets would have been helpful in this regard. It is the author's intent to downplay some of the formality and to place a greater emphasis instead on examples and problems. Indeed, the exercises and examples play an integral role in this text, and they should not be overlooked because they are often sources of definitions and results that are invoked elsewhere (at times without reference). This is more frequently the case in the sections on groups and somewhat less pronounced in the sections on rings and fields. For example, Sylow's theorem for abelian groups is an unlabelled exercise, but the first mention of Sylow's p-subgroup comes eight additional pages later in a rather off-hand manner, as if to imply that the reader already knows what such a subgroup is. The definition of Sylow's $p$-subgroup is given seven pages after that. The proof of the theorem that the symmetric group $S$, on $n$ letters, has the alternating group as its only nontrivial normal subgroup when $n ≥ 5$ uses, without reference, the fact that the centre of $S$ is trivial. The exercise to demonstrate the triviality of the centre follows a few pages later. Most of these logistical problems could easily be avoided by a few well-placed references, eliminating some of the difficulties they now create for the instructor and student using the book for the first time.

There are also several places in the text where some model methods and problems would have been a great help. It is noted that by direct examination we discover the greatest common divisor of 24 and 9 is 3, but since the Euclidean algorithm is not presented for polynomials or for integers, the reader is left to wonder if every greatest common divisor must be so divined. The useful formula for computing the product $\tau \sigma \tau^{-1}$ for $\tau, \sigma$ in the symmetric group is relegated to the special topics section at the end, where it might be omitted for lack of time. Such concrete calculations would nicely complement the text material, have a practical value, and afford a respite from the abstract theory.

These points aside, it should be stated that throughout the text, care is taken to achieve interesting, applicable, and significant results about each of the fundamental algebraic systems studied - groups, rings, and fields. Constructibility and the special topics at the end, especially the irrationality of π, provide entertaining sidelights to the main topics. A real attempt has been made to prepare the ground for new concepts, to give many varied examples, and to present a wide range of challenging problems (662, by the author's count). There is more than enough material for a one-semester course. The material on fields could be postponed for a second course, where it could be supplemented with topics from linear algebra such as matrices, vector spaces, transformations, and bilinear forms. No background in calculus or linear algebra is assumed, but certainly the level of sophistication of the text and its problems demands much mathematical maturity from its readers
...
Abstract Algebra has much to offer in its readability and clarity. Both the author and the publisher, Macmillan, should be commended on that score. The examples and exercises enhance and enrich the fine exposition and are valuable resources in themselves. There are a few logistical difficulties which undoubtedly will be ironed out by the time the second edition rolls off the press, as I expect this Herstein book, too, will be around a long time to influence a new generation of mathematicians and another generation of textbooks.

9.3. Review by: Frederick Hoffman.
Mathematical Reviews MR1011035 (92m:00002).

The "landmark" abstract algebra texts that remain in print certainly include van der Waerden's Algebra, Birkhoff-Mac Lane's Survey of modern algebra, and the author's Topics in algebra. Each has had multiple editions, and has illustrated trends in algebra and pedagogy. The treatment of transfinite mathematics in the various editions of van der Waerden, and the deletion of the adjective "moderne" from late editions are noteworthy, for example, as are the emergence of not only a briefer Birkhoff-Mac Lane survey, but also a Mac Lane-Birkhoff categorical alternative. The author's Topics in algebra is a beautiful book, which captured a large market, and became the text for almost everyone's ideal undergraduate course, in addition to making its author's name an adjective for graduate qualifying-exam-level algebra at many institutions. If one approached Topics in algebra in a manner consistent with its author's approach to the undergraduate course, that is, that the precise material covered, and the amount of it, was not as important as the way it was covered, then the book could be used almost anywhere, with the number of pages covered and percent of problems done a function of the audience and the instructor. The charm of the writing, and the wonderful exercises help it retain its position with many of us. A few years ago, though, the author decided to write another undergraduate algebra text, and it is the subject of this review.

The author intends to introduce the beginner to the wonderful world of abstract algebra, and to present enough material to make the introduction meaningful. The style is chatty, and there are enough exercises to provide ample drill, to expand coverage of topics that are treated in the text proper, and to challenge the strongest students. These are all features the present text shares with Topics in algebra, but there are many differences, the main being the reduced scope and slightly reduced starting level of the present volume. There are some attempts to "modernise". This includes moving functions to the left, so that composition goes from right to left. There is also the unfortunate change in names of cosets of subgroups: when the subgroup sits to the left of the representative, the product is called a "left coset"; something no one seems to do anymore.
...
What we have here is a better than average text for a first course in abstract algebra, having some of the charm of a Herstein book, but not of the class of Topics in algebra. It needed better proof-reading and copy editing, and also some work on the mathematics, since it clearly was released without proper care. In fact, the quantity of errors is sufficient to advise avoidance of the book for self-study, although text use, with a careful instructor, is reasonable.

10.1 From the Publisher.

Matrix Theory and Linear Algebra by Israel N Herstein is a comprehensive textbook that covers the fundamental concepts and applications of linear algebra and matrix theory. The book is designed to provide a solid foundation in these topics for undergraduate and graduate students studying mathematics, engineering, computer science, and other related disciplines. It begins with an introduction to basic matrix operations, including addition, subtraction, multiplication, and inversion, and then progresses to more advanced topics such as determinants, eigenvalues, eigenvectors, and diagonalization. The text also explores applications of linear algebra in areas such as systems of linear equations, linear transformations, and vector spaces. Herstein's clear and concise writing style, along with numerous examples and exercises, make this book an invaluable resource for students seeking to enhance their understanding of linear algebra and matrix theory. Whether used as a primary textbook or a supplementary resource, Matrix Theory and Linear Algebra provides a solid theoretical foundation and practical skills that are essential for success in advanced mathematics and related fields.

10.2. From the Contents.

1. The 2 × 2 Matrices
2. Systems of Linear Equations
3. The $n \times n$ Matrices
4. More on $n \times n$ Matrices
5. Determinants
6. Rectangular Matrices
7. More on Systems of Linear Equations
8. Abstract Vector Spaces
9. Linear Transformations
10. The Jordan Canonical Form Optional

11. EQUATIONS
12. Applications Optional
13. Least Squares Methods Optional
14. Linear Algorithms Optional

Index

10.3. Review by: Sándor Lajos.
Mathematical Reviews MR0999134 (90h:15001).

This is an interesting linear algebra textbook including determinants and linear transformations, the Cayley-Hamilton theorem, inner product spaces, the Gram-Schmidt process, least square methods, and applications (Markov processes, differential equations, etc.). The authors discuss how to translate some of the methods into algorithms. There are illustrations of these algorithms used in computer programs (written in the programming language Pascal). There are many good exercises in the book. This reviewer can recommend this book for any one- or two-semester linear algebra course on an introductory or advanced level.

11.1. From the Preface by David J Winter.

When I was invited to prepare this Second Edition of Herstein's Abstract Algebra, I felt that it would be a mistake to undertake any major changes in such an attractive and well balanced treatment of the subject. This opinion was echoed by everyone with whom I conferred. So, no major changes have been made, and Herstein's inimitable style and choice of content remain virtually untouched. At the same time, important minor changes thread their way throughout the Second Edition. Some fix minor errors and rough spots, and others expand and clarify the discussion and examples. There are also two small changes in format.

First, a Symbol List has been added, for readers to use when they confront a forgotten symbol or look ahead.

Second, and more importantly, a few problems have been marked with an asterisk (*).

These problems serve as a vehicle to introduce some concepts and simple arguments which support or relate in some interesting way to the discussion. As such, they should be read carefully.

I am grateful to several people whose very substantial contributions to this edition will help make this delightful book by Herstein even more enjoyable for the reader. I take this opportunity to thank Georgia Benkart, Barbara Cortzen and Lynne Small for their creative and highly useful suggestions.

11.2. Review by: Frederick Hoffman.
Mathematical Reviews MR1114754 (92m:00003).

It was clear that a second edition was needed, to smooth out the rough spots in the hastily produced first edition. According to the preface, Winter undertakes to "leave Herstein's inimitable style and choice of content virtually unchanged", but to "fix minor errors and rough spots, and to expand and clarify the discussion and examples". He definitely adhered to the first commitment. The repair, expansion and clarification job is, in the opinion of the reviewer, done poorly, and Herstein is not well served by either Professor Winter or the publisher. He and we the readers deserved better, and it would be dishonest for this reviewer to say otherwise. I shall limit myself to some general comments since to elaborate on the most serious flaws remaining from the first edition (1986) and the new ones which have been introduced in the second edition would require a far too lengthy review.

When new material is added, to amplify or clarify existing material, it is generally done in a poor way. There are cases of different type sizes, blurred type, lost words and symbols, new typos, and in one brilliant example to prove that no one looked hard at the finished product, a footnote has migrated to the middle of the page after the one it references. As far as the accuracy of the error-correction, I estimate that no more than half the errors were picked up, and some new ones were added. With respect to the needs for improvement of the exposition, this is more subjective, but I would definitely rate the job as sub-standard. In summary the promising but somewhat flawed first edition deserved a carefully edited, including copy edited, second edition which, unfortunately, it did not receive.

12.1. From the preface by Barbara Cortzen and David J Winter.

When we were asked to prepare the third edition of this book, it was our consensus that it should not be altered in any significant way, and that Herstein's informal style should be preserved. We feel that one of the book's virtues is the fact that it covers a big chunk of abstract algebra in a condensed and interesting way. At the same time, without trivialising the subject, it remains accessible to most undergraduates.

We have, however, corrected minor errors, straightened out inconsistencies, clarified and expanded some proofs, and added a few examples. To resolve the many typographical problems of the second edition, Prentice Hall has had the book completely re-typeset - making it easier and more pleasurable to read.

It has been pointed out to us that some instructors would find it useful to have the Symmetric Group S_n and the cycle notation available in Chapter 2, in order to provide more examples of groups. Rather than alter the arrangement of the contents, thereby disturbing the original balance, we suggest an alternate route through the material, which addresses this concern. After Section 2.5, one could spend an hour discussing permutations and their cycle decomposition (Sections 3.1 and 3.2), leaving the proofs until later. The students might then go over several past examples of finite groups and explicitly set up isomorphisms with subgroups of $S_{n}$. This exercise would be motivated by Cayley's theorem, quoted in Section 2.5. At the same time, it would have the beneficial result of making the students more comfortable with the concept of an isomorphism. The instructor could then weave in the various subgroups of the Symmetric Groups $SL_{n}$ as examples throughout the remainder of Chapter 2. If desired, one could even introduce Sections 3.1 and 3.2 after Section 2.3 or 2.4.

Two changes in the format have been made since the first edition. First, a Symbol List has been included to facilitate keeping track of terminology. Second, a few problems have been marked with an asterisk (*). These serve as a vehicle to introduce concepts and simple arguments that relate in some important way to the discussion. As such, they should be read carefully.

Finally, we take this opportunity to thank the many individuals whose collective efforts have helped to improve this edition. We thank the reviewers: Kwangil Koh from North Carolina State University, Donald Passman from the University of Wisconsin, and Robert Zinc from Purdue University. And, of course, we thank George Lobell and Elaine Wetterau, and others at Prentice Hall who have been most helpful.