Symmetry: An Introduction to Group Theory and Its Applications (Dover Books on Physics)

The author states in the preface that this book was written to serve readers at a variety of levels. In fact this rather short book is almost three books in one. The first two chapters provide an introduction to groups, vector spaces, and lattices. The material here is elementary, but the author is to be commended both for clear explanations as well as excellent notation. My only complaint is that the notation is useful to read, but not really to write as it relies on different typefaces. The next two chapters dig into translation groups, point groups, and space groups in three dimensions. Chapter 5 focuses on the theory of irreducible representations, and in many ways forms the core of the book. The material in these chapters is definitely more challenging than the material in the first two chapters. The final three chapters are devoted to applications. Chapter 6 covers applications to algebraic forms particularly as applied to vibrational modes as encountered in spectroscopy. Chapter 7 focuses on applications to functions and operators. Finally, chapter 8 is devoted to tensors and tensor operators. These three chapters are the most advanced in the book, and each is successively more advanced than the preceding one. These chapters will serve more advanced readers. Early on the author introduces the group which comes to be known as C_3v. This groups is used as an example throughout the book and comes fully to life in chapter 6 where it is used to analyze the vibrational modes of the ammonia molecule. I really appreciated the continuity of and the elaboration on this one example as a unifying thread in the text. The entire book contains copious tables and figures which are extremely helpful. In particular, chapters 3 and 4 contain detailed tables of the groups considered as well as their representations. The book has no exercises, but does offer a nice collection of suggestions for further reading at the end of each chapter. Finally, this book has remarkably few errors which is actually somewhat surprising just given the level of detail in some sections such as the group tables mentioned above. Sure to be useful in particular to students of quantum chemistry and spectroscopy, this book is excellent and repays multiple readings, but I'd recommend the prospective reader bring a background in linear algebra and some quantum chemistry.

This is one of the finest examples of didactic mathematics I have ever seen. Good teaching is rare in mathematics...sorry, I wish it weren't so, but mathematicians often suffer from profound cluelessness about what is obvious and what is a stumbling block to understanding. Not so McWeeney, who understands the didactic principle of a continuing example that is developed and built upon along with the theory (here a simple rotation group of the triangular lamina, i.e. turning and flipping a triangle cutout). And no super-challenging problem sets and unfinished proofs that just make the math mortal feel stupid and want to give up. This presentation could take any avid high school Algebra II student well into group theory and matrix representations. That said, one would be better off with a background of basic matrix theory such as identity matrix, inverse matrix, etc., and some comfort with row/column language and the dual subscripts of matrices. But even that would not be necessary for anyone who comprehends sigma notation of sums and the basic principles of subscripting/indexing. The first chapter is harder by far than the second. Readers should just remember that fraction groups are like riding a bike; once seen for what they are it will all come into place. If you need a companion to this you might consider Byron and Fuller's Dover classic text, which develops vectors even more rigorously and from a more numeric perspective.

I wanted to learn about symmetry for use in understanding the quantum understanding of the periodic table. The orbital groups, labeled with the letters S, P, D, and F seem to involve symmetry. They are often said to be solutions to the Schrodinger equation. However this book gives a mathematical description of symmetry groups and a geometric way of thinking about including and excluding various possibilities as solutions. The pragmatics seem very powerful. This book is designed for mathematicians. Since I am not and I am only interested in learning and teaching about the periodic table, I skipped over the mathematical notation’s and was able to extract principles of group theory and symmetry as it pertains to the periodic table. Therefore I found the book very valuable.

I read the first four chapters of this book to get up to speed on group theory and symmetry because I needed the knowledge to write a peer reviewed paper in crystallography (original research). Other than reading a short introduction on-line, my knowledge of group theory before reading this book could be boiled down to one sentence: "There is something called 'group theory'." I literally didn't even realize that the "group" in "symmetry group" related to group theory. The point is, this book had everything I needed to turn my vague ideas about symmetry in protein secondary structure into a cohesive group theoretic treatment and formal proof. After re-formulating my paper using the concepts presented in this book, I was able to send my paper into one of the top crystallography journals with all referees enthusiastically recommending publication. Google "The Zipper Groups of the Amyloid State of Proteins" (it's an open access article). What's more, this book had much more about symmetry than I needed. I stopped reading before the chapters on chemistry (5-8) because of time. I hope one day to go back and read those chapters because I am sure that this book will benefit my understanding of chemistry as much as it has my understanding of symmetry.

Wondering what a coset is? How axioms for lattices and groups generalize to vector spaces and linear algebra. Like an example of symmetry used as a theme for the whole book? This is the book for you.

I can see why this book has positive reviews for its presentation. A good teacher starts with what the students understand, the intuitions they have developed, a common point of understanding and then motivates then to take a journey from this point to areas and topics that the student does not know nor understand ... yet. The author starts with counting and uses informal presentations and examples to present concepts of Linear Algebra, Abstract Algebra, Group Theory, and Representation Theory, always anchored in the physical reality of understanding Symmetry of physical things such as atoms, molecules, and crystals. I must say having worked with several other text books on group theory and many presentations on application to chemistry and crystallography the journey here is not unpleasant, though the informality has sort of tap down my urges to calculate things since most things have yet to be formally defined through chapter 3. But by the end of chapter 3 I already know I am in a room of no exit. The material covered in Chapter 3 is quite astonishing. Instead of dealing initially with point symmetries, the author starts with space symmetries that include point rotations and reflections as well as translations, with a taste of representation theory, a embryonic character table, and a smidge of unitary transforms, we are covering all the generators for all point symmetries of degree 1, 2, 3, 4, and 6 as well as polyhedron groups T and O. This leads to an excellent presentation of crystal groups, which I have not seen so concisely presented before, and a proof that this is all that is necessary to understand to apply to physical symmetries that include translation (I.e. crystallography). The presentation of Icosahedron symmetries and the introduction of quasi-cyrstals resulting from these symmetries is not included which breaks from the conventional wisdom presented here. For someone who wants or needs to come up to speed on typical applications of group theory to chemistry and physics this may not be so bad. But I am currently focused on polyhedral symmetries and their representations, this another dead end. I actually appreciate that the book explains well the conventional wisdom that permeates the development and its applications in chemistry and crystallography, though most of the actual concepts and theorems have yet to be defined and discussed. This leads me to my initial point, I have been lead to this point by my intuitive understanding - I believe that I understand what the author is presenting but it is not clear yet that I can do anything with that understanding. I will have to see as I proceed through the book if I will discovery actionable knowledge. . I have some time before I receive my next text book on group theory.

This book is fantastic! We dont have any applied algebra text so clear like this book in portuguese. Good for algebra teachers who want to develop higher topics with students.

The only thing I'd ask for is larger print. The book is interesting, understandable and a nice follow up to David Nash's book on group theory.

Los textos sobre la materia acostumbran a ser una especie de libros de leyes donde te vienen a enumerar las propiedades, ecuaciones, etc, demostrándotelas desde un punto de vista puramente formal, sin conexión alguna con la realidad. Este libro es un prodigio de dicáctica, sin perder la profundidad necesaria, y está lleno de ideas, de ideas normales escritas con palabras normales en inglés. Si alguna persona está buscando un libro que le explique bien que es la Teoría de Grupos pienso que éste es el texto. Para su lectura es necesaria una base mínima en álgebra y en las cuestiones básicas de las matemáticas.

very pleased with every thing