By many expert mathematicians, group theory is often addressed as a central part of mathematics. It finds its origins in geometry, since geometry describes groups in a detailed manner. The theory of polynomial equations also describes the procedure and principals of associating a finite group with any polynomial equation. This association is done in such a way that makes the group to encode information that can be used to solve the equations. This equation theory was developed by Galois. Finite group theory faced a number of changes in near past times as a result of classification of finite simple groups. The most important theorem when practicing group theory is theorem by Jordan holder. This theorem shows how any finite group is a combination of multiple finite simple groups. Group theory is a term that is mainly used fields related to mathematics such as algebraic calculations. In abstract algebra, groups are referred as algebraic structures. Other terms of algebraic theories, such as rings, fields and vector spaces are also seen as group. Of course with some additional operations and axioms, mathematicians accept them as a group. The methods and procedures of group theory effect many parts and concepts of mathematics as well as algebra on a large scale. Linear algebraic groups and lie groups are two main branches or say categories of group theory that have advanced enough to be considered as a subject in their own perspectives.

The Mathematical Study Of Group Theory Was Initiated In The Early Nineteenth Century By Such Mathematicians As Gauss, Cauchy, Abel, Hamilton, Galois, Cayley, And Many Others. However, The Advantages Of Group Theory In Physics Were Not Recognized Till 1925 When It Was Applied For Formal Study Of Theoretical Foundations Of Quantum Mechanics, Atomic Structures And Spectra By, To Name A Few, H A Bethe, E P Wigner, Etc. It Has Now Become Indispensable In Several Branches Of Physics And Physical Chemistry.Dr. Joshi Develops The Mathematics Of Group Theory And Then Goes On To Present Its Applications To Quantum Mechanics, Crystallography, And Solid State Physics. For Proper Comprehension Of Representation Theory, He Has Covered Thoroughly Such Diverse But Relevant Topics As Hilbert Spaces, Function Spaces, Operators, And Direct Sum And Product Of Matrices. He Often Proceeds From The Particular To The General So That The Beginning Student Does Not Have An Impression That Group Theory Is Merely A Branch Of Abstract Mathematics. Various Concepts Have Been Explained Consistently By The Use Of The C4V. Besides, It Contains An Improved And More General Proof Of The Schurs First Lemma And An Interpretation Of The Orthogonality Theorem In The Language Of Vector Spaces (Chapter 3).Throughout The Text The Author Gives Attention To Details And Avoids Complicated Notation. This Is A Valuable Book For Senior Students And Researchers In Physics And Physical Chemistry. A Thorough Understanding Of The Methodology And Results Contained In This Book Will Provide The Reader Sound Theoretical Foundations For Advanced Study Of Quantum Mechanics, Solid State Physics And Atomic And Particle Physics To Help Students A Flow-Chart Explaining Step By Step The Method Of Determining A Parallel-Running Example Illustrating The Procedure In Full Details Have Been Included. An Appendix On Mappings And Functions Has Also Been Added.

This textbook provides a readable account of the examples and fundamental results of groups from a theoretical and geometrical point of view. Topics on important examples of groups (like cyclic groups, permutation groups, group of arithmetical functions, matrix groups and linear groups), Lagrange’s theorem, normal subgroups, factor groups, derived subgroup, homomorphism, isomorphism and automorphism of groups have been discussed in depth. Covering all major topics, this book is targeted to undergraduate students of mathematics with no prerequisite knowledge of the discussed topics. Each section ends with a set of worked-out problems and supplementary exercises to challenge the knowledge and ability of the reader.

This book is addressed to graduate students and research workers in theoretical physics who want a thorough introduction to group theory and Hopf algebras. It is suitable for a one-semester course in group theory or a two-semester course which also treats advanced topics. Starting from basic definitions, it goes on to treat both finite and Lie groups as well as Hopf algebras. Because of the diversity in the choice of topics, which does not place undue emphasis on finite or Lie groups, it should be useful to physicists working in many branches. A unique aspect of the book is its treatment of Hopf algebras in a form accessible to physicists. Hopf algebras are generalizations of groups and their concepts are acquiring importance in the treatment of conformal field theories, noncommutative spacetimes, topological quantum computation and other important domains of investigation. But there is a scarcity of treatments of Hopf algebras at a level and in a manner that physicists are comfortable with. This book addresses this need superbly. There are illustrative examples from physics scattered throughout the book and in its set of problems. It also has a good bibliography. These features should enhance its value to readers. The authors are senior physicists with considerable research and teaching experience in diverse aspects of fundamental physics. The book, being the outcome of their combined efforts, stands testament to their knowledge and pedagogical skills.

While group theory and its application to solid state physics is well established, this textbook raises two completely new aspects. First, it provides a better understanding by focusing on problem solving and making extensive use of Mathematica tools to visualize the concepts. Second, it offers a new tool for the photonics community by transferring the concepts of group theory and its application to photonic crystals. Clearly divided into three parts, the first provides the basics of group theory. Even at this stage, the authors go beyond the widely used standard examples to show the broad field of applications. Part II is devoted to applications in condensed matter physics, i.e. the electronic structure of materials. Combining the application of the computer algebra system Mathematica with pen and paper derivations leads to a better and faster understanding. The exhaustive discussion shows that the basics of group theory can also be applied to a totally different field, as seen in Part III. Here, photonic applications are discussed in parallel to the electronic case, with the focus on photonic crystals in two and three dimensions, as well as being partially expanded to other problems in the field of photonics. The authors have developed Mathematica package GTPack which is available for download from the book's homepage. Analytic considerations, numerical calculations and visualization are carried out using the same software. While the use of the Mathematica tools are demonstrated on elementary examples, they can equally be applied to more complicated tasks resulting from the reader's own research.

This book presents articles at the interface of two active areas of research: classical topology and the relatively new field of geometric group theory. It includes two long survey articles, one on proofs of the Farrell–Jones conjectures, and the other on ends of spaces and groups. In 2010–2011, Ohio State University (OSU) hosted a special year in topology and geometric group theory. Over the course of the year, there were seminars, workshops, short weekend conferences, and a major conference out of which this book resulted. Four other research articles complement these surveys, making this book ideal for graduate students and established mathematicians interested in entering this area of research.

A comprehensive discussion of group theory in the context of molecular and crystal symmetry, this book covers both point-group and space-group symmetries. - Provides a comprehensive discussion of group theory in the context of molecular and crystal symmetry - Covers both point-group and space-group symmetries - Includes tutorial solutions

The Application of Group Theory in Physics is a 17-chapter text based on a course of lectures concerning the principles, concepts, and application of group theory in physics, given at the Gorki University in Kharkov. This text presents first the parts of the theory of representations of finite and continuous groups that are most important in application. Considerable chapters cover the groups of theory of interest in theoretical physics and demonstrate the principles according to which the abstract concepts and the theorems of representation theory are applied in theoretical physics. The remaining chapters provide representations of the rotation group and the Lorentz group. The closing part of this work contains tables of the detailed description of the 230 space groups and for the characters of certain groups. This book is intended primarily for physicists specializing in theoretical physics