Customer Reviews

Groups and Symmetry (Undergraduate Texts in Mathematics)

5 star:		(3)
4 star:		(2)
3 star:		(0)
2 star:		(1)
1 star:		(0)

 

 

This was the textbook for my first course in abstract algebra and the first "yellow book" that I read. I found it an excellent book: rather than starting with axioms and dryly deriving everything, it gets one to contemplate the meaning and motivation behind the axioms. This book will encourage you to play around with mathematics on paper and in your mind, helping you to get a concrete feel for a subject that many people view as painfully abstract.

The prose is clear and well-written: there is just the right amount of discussion to elucidate necessary points, while allowing the book to remain fairly compact. Exercises are fun but difficult and many require genuine creativity.

I also really like the choice of topics: although this book is introductory (with respect to abstract algebra, it presupposes some knowledge of linear algebra), because it focuses only on groups (as opposed to also trying to handle rings & fields) it is able to get into some more advanced and very interesting topics and applications in later chapters. This book will give you a lot more than can be covered in a single semester undergrad course, and while it doesn't exactly make the best reference text, it will be a book you will want to keep coming back to, if only to study some of the more advanced material.

There are differing perspectives on the teaching of abstract algebra: some people like to start with group theory exclusively in a first course, and treat rings, fields, and other structures in later courses. Other people recommend more integrated approaches, or approaches starting from rings. While I can't say that either approach is better, I can say that this book takes the first approach, focusing exclusively on groups and assuming little prior background..and for a first course in abstract algebra, this book is an excellent choice.

Please note that the other reviews here are obviously for some other book. This is not an advanced text on bifurcations and stability. It is an introductory book on group theory. I have been using this book for self study. It is well suited to this purpose. The book uses symmetry to unify and motivate the study of groups. The discussion of the symmetry groups of Platonic solids is both enjoyable in itself and useful for visualizing groups. The chapters are very short. The exercises are well suited to gaining insight into the material.

I'm using this book for a first course in Group theory and it makes a good introductory text. Topics are neatly arranged and follow an order that makes reading easy. But it is definitely not a text for any sort of rigorous proofs. Rather, it focuses on learning from examples.

Some authors like to expound, some like to dazzle and some like to teach. Armstrong falls easily into the latter category. If you are looking for a clear and motivating book to start learning about group theory, you will not find a better book. It is a short book but covers the essentials. The range of topics covers the necessary ground for an introduction: Sylow's theorems, free groups, matrix groups, presentations are all there with a strong geometric content. He even proves the Nielsen-Schrier Theorem in an accessible manner. All you really need to know is some basic undergraduate algebra to understand this little gem and it will certainly give you the foundation to move deeper. Highly recommended for a starting point.

The book consists solely of exercises and hints for every exercise, which he curiously calls "answers". This book is perfect if you are looking to review geometrically-tinged algebraic structures like matrix groups, symmetry groups, and wallpaper groups. There is also some basic pure algebra in here. I don't think this book would work all that well for a student new to algebra, although someone with some backgroud in algebra can definitely get something out of the geometric chapters.

Armstrong is ridiculously laconic. He also writes as if you're fluent in the language of abstract pure mathematics. He also assumes you remember theorems proved in homework exercises and uses them in the main text of later chapters. Do yourselves a favor and take an intermediate level algebra course before picking up this book (if you even have to pick up this book, which I wouldn't have, but it was required for my course). Applications happen on a rare basis, until the last handful of chapters. Worse still, Armstrong decides to only allude (at most) to scientific applications of the material, and instead focuses the "applied" chapters on things such as block problems (how many blocks can be produced if you have two colors of paint and paint pattern x) and wallpaper patterns (whereas he could have focused on the numerous applications of group theory to chemistry and even physics). Don't let the easy going and easy to understand first 3 or 4 chapters fool you, this is really a poor text except perhaps for those who already know group theory, know high level algebra, and just want a book explaining the intersections in a not-so-rigorous, often hand-wavy way (this also adds to frustration because your professor will probably tell you your proof is weak if you mimic the proofs in the chapter text).