Dinesh Khattar Neha Agrawal Group Theory

This textbook is designed for the UG/PG students of mathematics for all universities over the world. It is primarily based on the classroom lectures, the authors gave at the University of Delhi. This book is used both for self-study and course text. Full details of all proofs are included along with innumerous solved problems, interspersed throughout the text and at places where they naturally arise, to understand abstract notions. The proofs are precise and complete, backed up by chapter end problems, with just the right level of difficulty, without compromising the rigor of the subject. The book starts with definition and examples of Rings and logically follows to cover Properties of Rings, Subrings, Fields, Characteristic of a Ring, Ideals, Integral Domains, Factor Rings, Prime Ideals, Maximal Ideals and Primary Ideals, Ring Homomorphisms and Isomorphisms, Polynomial Rings, Factorization of Polynomials, and Divisibility in Integral Domains.

The book is intended to serve as an introductory course in group theory geared towards second-year university students. It aims to provide them with the background needed to pursue more advanced courses in algebra and to provide a rich source of examples and exercises. Studying group theory began in the late eighteenth century and is still gaining importance due to its applications in physics, chemistry, geometry, and many fields in mathematics.

The book is intended to serve as an introductory course in group theory geared towards second-year university students. It aims to provide them with the background needed to pursue more advanced courses in algebra and to provide a rich source of examples and exercises. Studying group theory began in the late eighteenth century and is still gaining importance due to its applications in physics, chemistry, geometry, and many fields in mathematics.

We are pleased to present the book “Geometric Group Theory, Dortmund and Carleton Conferences”, a selection of the best research articles from two strongly related 2007 international conferences:

This textbook provides a didactic introduction to the topic of group theory in physics, with a special focus on solid state physics issues. The book is useful for students who encounter such problems in their first scientific work (in theory or experiment). In addition to the basic introduction to group theory and representation theory, the book deals with point groups, double point groups, and space groups, which are essential in solid state physics. As an example for systems with space group symmetry, electrons in periodic potentials are discussed.

This book is based on a one-semester course for advanced undergraduates specializing in physical chemistry. I am aware that the mathematical training of most science majors is more heavily weighted towards analysis – typ- ally calculus and differential equations – than towards algebra. But it remains my conviction that the basic ideas and applications of group theory are not only vital, but not dif?cult to learn, even though a formal mathematical setting with emphasis on rigor and completeness is not the place where most chemists would feel most comfortable in learning them.

This volume is a consequence of a series of seminars presented by the authors at the Infrared Spectroscopy Institute, Canisius College, Buffalo, New York, over the last nine years. Many participants on an intermediate level lacked a sufficient background in mathematics and quantum mechan­ ics, and it became evident that a non mathematical or nearly nonmathe­ matical approach would be necessary. The lectures were designed to fill this need and proved very successful. As a result of the interest that was developed in this approach, it was decided to write this book. The text is intended for scientists and students with only limited theore­ tical background in spectroscopy, but who are sincerely interested in the interpretation of molecular spectra. The book develops the detailed selection rules for fundamentals, combinations, and overtones for molecules in several point groups. Detailed procedures used in carrying out the normal coordinate treatment for several molecules are also presented. Numerous examples from the literature illustrate the use of group theory in the in­ terpretation of molecular spectra and in the determination of molecular structure.

Every process in physics is governed by selection rules that are the consequence of symmetry requirements. The beauty and strength of group theory resides in the transformation of many complex symmetry operations into a very simple linear algebra. This concise and class-tested book has been pedagogically tailored over 30 years MIT and 2 years at the University Federal of Minas Gerais (UFMG) in Brazil. The approach centers on the conviction that teaching group theory in close connection with applications helps students to learn, understand and use it for their own needs. For this reason, the theoretical background is confined to the first 4 introductory chapters (6-8 classroom hours). From there, each chapter develops new theory while introducing applications so that the students can best retain new concepts, build on concepts learned the previous week, and see interrelations between topics as presented. Essential problem sets between the chapters also aid the retention of the new material and for the consolidation of material learned in previous chapters. The text and problem sets have proved a useful springboard for the application of the basic material presented here to topics in semiconductor physics, and the physics of carbon-based nanostructures.

In recent years, many students have been introduced to topology in high school mathematics. Having met the Mobius band, the seven bridges of Konigsberg, Euler's polyhedron formula, and knots, the student is led to expect that these picturesque ideas will come to full flower in university topology courses. What a disappointment "undergraduate topology" proves to be! In most institutions it is either a service course for analysts, on abstract spaces, or else an introduction to homological algebra in which the only geometric activity is the completion of commutative diagrams. Pictures are kept to a minimum, and at the end the student still does not understand the simplest topological facts, such as the reason why knots exist. In my opinion, a well-balanced introduction to topology should stress its intuitive geometric aspect, while admitting the legitimate interest that analysts and algebraists have in the subject. At any rate, this is the aim of the present book. In support of this view, I have followed the historical develop­ ment where practicable, since it clearly shows the influence of geometric thought at all stages. This is not to claim that topology received its main impetus from geometric recrea. ions like the seven bridges; rather, it resulted from the visualization of problems from other parts of mathematics­ complex analysis (Riemann), mechanics (poincare), and group theory (Oehn). It is these connections to other parts of mathematics which make topology an important as well as a beautiful subject.

In recent years, many students have been introduced to topology in high school mathematics. Having met the Mobius band, the seven bridges of Konigsberg, Euler's polyhedron formula, and knots, the student is led to expect that these picturesque ideas will come to full flower in university topology courses. What a disappointment "undergraduate topology" proves to be! In most institutions it is either a service course for analysts, on abstract spaces, or else an introduction to homological algebra in which the only geometric activity is the completion of commutative diagrams. Pictures are kept to a minimum, and at the end the student still does not understand the simplest topological facts, such as the reason why knots exist. In my opinion, a well-balanced introduction to topology should stress its intuitive geometric aspect, while admitting the legitimate interest that analysts and algebraists have in the subject. At any rate, this is the aim of the present book. In support of this view, I have followed the historical develop­ ment where practicable, since it clearly shows the influence of geometric thought at all stages. This is not to claim that topology received its main impetus from geometric recrea. ions like the seven bridges; rather, it resulted from the visualization of problems from other parts of mathematics­ complex analysis (Riemann), mechanics (poincare), and group theory (Oehn). It is these connections to other parts of mathematics which make topology an important as well as a beautiful subject.