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Customer Reviews (1)

A useful guide
Short, and much more accessible than other modern treatments, getting quickly to the meat and bones. Loewner eschews "abstraction for the sake of abstraction", which lets you see what the content of continuous groups is about (representations, integration on groups) without your getting lost in a maze of formalism: you feel rooted.
To get through this little book quickly, you need to have a sense of what cosets do for you (i.e., not bothering to distinguish between elements that are equivalent to each other), and the notion in linear algebra of decomposing linear transformations orthogonally.
Loewner is especially useful if you've decided to dip into the three-volume work by Lie/Engels, or (better) the introductory version worked out by Scheffers of Lie's Vorlesungen ueber continuierliche Gruppen. Loewner's lectures in fact result from his attempt to get something out of the Lie/Engel treatise. The difficulty with trying to read Lie is that while he thought in terms of a group as an element in an r-dimensional sub-manifold of a transformation space, never-the-less he felt he had to couch his presentation in a form more congenial to the analysts of his day, which obscured his presentation. He added in small print what he was really trying to do -- structurally, and even with commutative diagrams! -- but first you have to wade through the analytic component-by-component treatment. What Loewner did was to extract the essence for you.
I recommend Loewner's precis, along with the more leisurely introduction by Lie/Scheffers. ... Read more

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Group Theory is an indispensable mathematical tool in many branches of physics and chemistry.Providing a detailed account of Group Theory and its applications to chemical physics, the first half of this book focuses on elementary topics, such as molecular and crystal symmetry. The latter, more advanced portion includes proven examples.The book's comprehensive coverage makes it suitable for a two-semester course at the advanced undergraduate and graduate levels. ... Read more

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This second edition develops the foundations of finite group theory.For students already exposed to a first course in algebra, it serves as a text for a course on finite groups.For the reader with some mathematical sophistication but limited knowledge of finite group theory, the book supplies the basic background necessary to begin to read journal articles in the field. It also provides the specialist in finite group theory with a reference on the foundations of the subject. Unifying themes include the Classification Theorem and the classical linear groups. Lie theory appears in chapters on Coxeter groups, root systems, buildings, and Tits systems. This second edition has been considerably improved with a completely rewritten Chapter 15 considering the 2-Signalizer Functor Theorem, and the addition of an appendix containing solutions to exercises. ... Read more

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This concise, class-tested book was refined over the authors’ 30 years as instructors at MIT and the University Federal of Minas Gerais (UFMG) in Brazil. The approach centers on the conviction that teaching group theory along with applications helps students to learn, understand and use it for their own needs. Thus, the theoretical background is confined to introductory chapters. Subsequent chapters develop new theory alongside applications so that students can retain new concepts, build on concepts already learned, and see interrelations between topics. Essential problem sets between chapters aid retention of new material and consolidate material learned in previous chapters.

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Building on the foundation of the Second Edition, Symmetry and Structure: Readable Group Theory for Chemists, Third Edition turns the complex and potentially difficult subject of group theory into an accessible and readable account of this core area of chemistry. By using a diagrammatical approach and demonstrating the physical principles involved in understanding group theory, the text provides a non-mathematical, yet thorough, treatment of this broad topic. This new edition has been fully revised and updated to include a much more three-dimensional and accurate visualization of many of the key topics. The chapter on octahedral molecules is extended to cover the important topic of the ligand field theory of octahedral transition metal complexes. Problems and summaries are included at the end of each chapter, the book provides detailed answers to frequently asked questions, and numerous diagrams and tables are featured for ease of reading and to enhance student understanding.

Symmetry and Structure: Readable Group Theory for Chemists, Third Edition is an essential textbook for all students, researchers and lecturers in chemistry, biochemistry, chemical engineering, physics and material science. ... Read more

Customer Reviews (2)

It's crap compared to Davison (which is out of print)
Okay so Davisons out of print but it's so much better (and called Group Theory For Chemists)!This book is a bit badly laid out, doesn't explain things very well and is pretty much useless unless you already have at least a simple grasp of Group Theory.I don't know what awful libraries you lot in Cleveland have but this is certainly not a particularly good book.

SFA Kettle: Brilliant teacher
In this text, Dr. Kettle presents the most lucent introduction availablefor students wishing to attain an understanding of Group Theory forchemical applications.The text is not overly mathematical in itspresentation, setting it apart from the numerous other texts available onthis subject.After having read this text, a student is amply able tounderstand discussions of group theory with respect to spectroscopy orotherwise. ... Read more

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-------------------Description--------------------The origins of computation group theory (CGT) date back to the late 19th and early 20th centuries. Since then, the field has flourished, particularly during the past 30 to 40 years, and today it remains a lively and active branch of mathematics.

The Handbook of Computational Group Theory offers the first complete treatment of all the fundamental methods and algorithms in CGT presented at a level accessible even to advanced undergraduate students. It develops the theory of algorithms in full detail and highlights the connections between the different aspects of CGT and with other areas of computer algebra. While acknowledging the importance of the complexity analysis of CGT algorithms, the authors' primary focus is on algorithms that perform well in practice rather than on those with the best theoretical complexity.

Throughout the book, applications of all the key topics and algorithms to areas both within and outside of mathematics demonstrate how CGT fits into the wider world of mathematics and science. The authors include detailed pseudocode for all of the fundamental algorithms, and provide detailed worked examples that bring the theorems and algorithms to life.---------------------Features---------------------· Provides the first reasonably accessible, comprehensive presentation of computational group theory· Summarizes state-of-the-art methods and results, including pointers to the literature for the principal areas of CGT· Incorporates the full underlying theory and correctness proofs of basic algorithms, and presents those algorithms in pseudocode· Includes a chapter on the precomputed stored libraries and databases of groups and character tables now publicly available---------------------Contents---------------------History of Computational Group Theory

BACKGROUND MATERIALSFundamentalsGroup ActionsSeriesPresentation of GroupsPresentation of SubgroupsAbelian Group PresentationsRepresentation Theory, Modules, Extension, Derivations, and ComplementsField Theory

REPRESENTING GROUPS ON A COMPUTERRepresenting Groups on ComputersThe Use of Random Methods in CGTSome Structural CalculatorsComputing with Homorphisms

COMPUTATION IN FINITE PERMUTATION GROUPSThe Calculation of Orbits and StabilizersTesting for Alt (W) and Sym (W)Finding Block SystemsBases and Strong Generating SetsHomomorphisms from Permutation GroupsBacktrack SearchesSylow Subgroups, P-cores, and the Solvable RadicalApplications

COSET ENUMERATIONThe Basic ProcedureStrategies for Coset EnumerationPresentations of SubgroupsFinding All SubgroupsFinding All Subgroups Up to a Given IndexApplications

PRESENTATION OF GIVEN GROUPSFinding a Presentation of a Given GroupFinding a Presentation of a Strong Generating SetThe Sims 'Verify' Algorithm

REPRESENTATIONS, COHOMOLOGY, AND CHARACTERSComputation in Finite FieldsElemetary Computational Linear AlgebraFactorizing Polynomials Over Finite FieldsTesting KG-Models for Irreducibility - The MeataxeRelated ComputationsCohomologyComputing Character TablesStructural Investigation of Matrix Groups

COMPUTATION WITH POLYCYCLIC GROUPSPolycyclic PresentationsExamples of Polycyclic GroupsSubgroups and Membership TestingFactor Groups and HomomorphismsSubgroup SeriesOrbit-Stabilizer MethodsComplements and ExtensionsIntersections, Centralizers, and NormalizersAutomorphism GroupsThe Structure of Finite Solvable Groups

COMPUTING QUOTIENTS OF FINITELY PRESENTED GROUPSFinite Quotients and Automorphism Groups of Finite GroupsAbelian QuotientsPractical Computation of the HNF and SNFP-Quotients of Finitely-Presented Groups

ADVANCED COMPUTATIONS IN FINITE GROUPSSome Useful SubgroupsComputing Composition and Chief SeriesApplications of the Solvable Radical MethodComputing the Subgroups of a Finite GroupAppication - Enumerating Finite Unlabelled Structures

LIBRARIES AND DATABASESPrimitive Permutation GroupsTransitive Permutation GroupsPerfect GroupsThe Small Groups LibraryCrystallorgraphic GroupsOther Databases

REWRITING SYSTEMSMonoid SystemsRewriting SystemsRewriting Systems in Monoids and GroupsRewriting Systems for Polycyclic GroupsVerifying NilpotencyApplications

FINITE STATE AUTOMATA AND AUTOMATIC GROUPSFinite State AutomataAutomatic GroupsThe Algorithm to Compute Shortlex Automatic StructuresRelated AlgorithmsApplications ... Read more

Customer Reviews (1)

Important Text on CGT
Handbook of Computational Group Theory by Derek F. Holt (Discrete Mathematics and Its Applications: Chapman & Hall/CRC) is about computational group theory, which we shall frequently abbreviate to CGT. The origins of this lively and active branch of mathematics can he traced back to the nineteenth and early twentieth centuries, but it has been flourishing particularly during the past 30 to 40 years. The aim of this book is to provide as complete a treatment as possible of all of the fundamental methods and algorithms in CGT, without straying above a level suitable for a beginning postgraduate student.
The most basic algorithms in CGT tend to be representation specific; that is, there are separate methods for groups given as permutation or matrix groups, groups defined by means of polycyclic presentations, and groups that are defined using a general finite presentation. The author has devoted separate chapters to algorithms that apply to groups in these different types of repre¬sentations, but there are other chapters that cover important methods involving more than one type. For example, Chapter 6 is about finding presentations of permutation groups and the connections between coset enumeration and methods for finding the order of a finite permutation group.
There is also included a chapter (Chapter 11) on the increasing number of precomputed stored libraries and databases of groups, character tables, etc. that are now publicly available. They have been playing a major rôle in CGT in recent years, both as an invaluable resource for the general mathematical public, and as components for use in some advanced algorithms in CGT. The library of all finite groups of order up to 2000 (except for order 1024) has proved to be particularly popular with the wider community.
It is inevitable that our choice of topics and treatment of the individual topics will reflect the authors' personal expertise and preferences to some extent. On the positive side, the final two chapters of the book cover appli¬cations of string-rewriting techniques to CGT (which is, however, treated in much greater detail, and the application of finite state automata to the computation of automatic structures of finitely presented groups. On the other hand, there may be some topics for which our treatment is more superficial than it would ideally be.
One such area is the complexity analysis of the algorithms of CGT. During the 1980s and 1990s some, for the most part friendly and respectful, rivalry developed between those whose research in CGT was principally directed to-wards producing better performance of their code, and those who were more interested in proving theoretical results concerning the complexity of the al¬gorithms. This study of complexity began with the work of Eugene Luks, who established a connection in his 1982 article between permutation group algorithms and the problem of testing two finite graphs for isomorphism. Our emphasis in this book will be more geared towards algorithms that per-form well in practice, rather than those with the best theoretical complexity. Fortunately, Seress' book includes a very thorough treatment of com¬plexity issues, and so we can safely refer the interested reader there. In any case, as machines become faster, computer memories larger, and bigger and bigger groups come within the range of practical computation, it is becom¬ing more and more the case that those algorithms with the more favourable complexity will also run faster when implemented.
The important topic of computational group representation theory and computations with group characters is perhaps not treated as thoroughly as it might be in this book. Some of the basic material is covered in Chapter 7, but there is unfortunately no specialized book on this topic.
One of the most active areas of research in CGT at the present time, both from the viewpoint of complexity and of practical performance, is the development of effective methods for computing with large finite groups of matrices. Much of this material is beyond the scope of this book. It is, in any case, developing and changing too rapidly to make it sensible to attempt to cover it properly here. Some pointers to the literature will of course be provided, mainly in Section 7.8.
Yet another topic that is beyond the scope of this book, but which is of increasing importance in CGT, is computational Lie theory. This includes computations with Coxeter groups, reflection groups, and groups of Lie type and their representations. It also connects with computations in Lie algebras, which is an area of independent importance. The article by Cohen, Murray, and Taylor provides a possible starting point for the interested reader.
The author firmly believes that the correct way to present a mathematical algorithm is by means of pseudocode, since a textual description will generally lack precision, and will usually involve rather vague instructions like "carry on in a similar manner". So we have included pseudocode for all of the most basic algorithms, and it is only for the more advanced procedures that we have occasionally lapsed into sketchy summaries. We are very grateful to Thomas Cormen who has made his LATEX package `clrscode' for displaying algorithms publicly available. This was used by him and his coauthors in the well-known textbook on algorithms.
Although working through all but the most trivial examples with procedures that are intended to be run on a computer can be very tedious, the author attempted to include illustrative examples for as many algorithms as is practical.
At the end of each chapter, or sometimes section, the reader's attention directed to some applications of the techniques developed in that chapter either to other areas of mathematics or to other sciences. It is generally difficult to do this effectively. Although there are many important and interesting applications of CGT around, the most significant of them will typically use methods of CGT as only one of many components, and so it not possible to do them full justice without venturing a long way outside of the main topic of the book.
The author assumes that the reader is familiar with group theory up to an advanced undergraduate level, and has a basic knowledge of other topics in algebra, such as ring and field theory. Chapter 2 includes a more or less complete survey of the required background material in group theory, but we shall assume that at least most of the topics reviewed will be already familiar to readers. Chapter 7 assumes some basic knowledge of group representation theory, such as the equivalence between matrix representations of a group G over a field K and KG-modules, but it is interesting to note that many of the most fundamental algorithms in the area, such as the `Meataxe', use only rather basic linear algebra.
... Read more

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Character theory is a powerful tool for understanding finite groups. In particular, the theory has been a key ingredient in the classification of finite simple groups. Characters are also of interest in their own right, and their properties are closely related to properties of the structure of the underlying group.The book begins by developing the module theory of complex group algebras. After the module-theoretic foundations are laid in the first chapter, the focus is primarily on characters. This enhances the accessibility of the material for students, which was a major consideration in the writing. Also with students in mind, a large number of problems are included, many of them quite challenging.In addition to the development of the basic theory (using a cleaner notation than previously), a number of more specialized topics are covered with accessible presentations. These include projective representations, the basics of the Schur index, irreducible character degrees and group structure, complex linear groups, exceptional characters, and a fairly extensive introduction to blocks and Brauer characters.This is a corrected reprint of the original 1976 version, later reprinted by Dover. Since 1976 it has become the standard reference for character theory, appearing in the bibliography of almost every research paper in the subject. It is largely self-contained, requiring of the reader only the most basic facts of linear algebra, group theory, Galois theory and ring and module theory. ... Read more

Customer Reviews (2)

An excellent book on character theory
This classic book offers one of the definite treatments of character theory of finite groups, written by one of the foremost experts in the field. The text assumes that the reader had been exposed to a one year sequence of graduate level algebra course (knowledge about the basics of the theory of finite groups is necessary to read this book; for example, the book "Groups and Representations" by Alperin and Bell provides the necessary background material).

The book is densely written and contains an incredible amount of material. The author provides concise but very readable proofs. Each chapter contains a large number of exercises that help the reader to master the material.

The enormous coverage of material makes this book one of the most useful references on (ordinary) characters of finite groups.

Not clear for beginners
Though Robert Greiss (of the Monster group fame) has assured me that the proofs and theory in the book are all sound, a beginner would have a difficult time believing it. The proofs generally leave a great deal to beproved, and worse to not point out these short comings so that a readercould at least work them out on the side. I am sure that if one understandsrepresentation theory then this is a marvelous reference, but for beginnersI prefer Curtis & Reiner's Methods of Representation Theory Volume 1,or even the last section of Dummit and Foote's Abstract Algebra which isthe best explained intro to representation and character theory that I havefound. ... Read more

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From the Preface: "From the 1950s until 1968, the theory of finite groups underwent an intense period of growth, including the first major classification theorem concerning simple groups as well as the construction of the first new sporadic simple group in a hundred years. In writing this book, my aim was to describe that development in sufficient detail for the interested reader to reach the frontiers of the subject and thereby participate in the excitement that then surrounded the study of simple groups ... In the intervening ten years [since the first edition of this work], an equally dramatic change has occurred ... a complete classification of the finite simple groups has now almost approached a reality ... Literally thousands of journal pages have been devoted to their study and powerful new techniques have been developed ... In most instances, these developments constitute a continuation rather than a replacement of the material in this book." ... Read more

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Combinatorial group theory is a loosely defined subject, with close connections to topology and logic. With surprising frequency, problems in a wide variety of disciplines, including differential equations, automorphic functions and geometry, have been distilled into explicit questions about groups, typically of the following kind: Are the groups in a given class finite (e.g., the Burnside problem)? Finitely generated? Finitely presented? What are the conjugates of a given element in a given group? What are the subgroups of that group? Is there an algorithm for deciding for every pair of groups in a given class whether they are isomorphic or not? The objective of combinatorial group theory is the systematic development of algebraic techniques to settle such questions.In view of the scope of the subject and the extraordinary variety of groups involved, it is not surprising that no really general theory exists. These notes, bridging the very beginning of the theory to new results and developments, are devoted to a number of topics in combinatorial group theory and serve as an introduction to the subject on the graduate level. ... Read more

Customer Reviews (2)

the best book in my life
The book is very instructive.

An interesting book about combinatorial group theory
This is an interesting book, dealing with subjects such as Burnside'sproblem, Nielsen-Schreier theorem, graphs of groups, amalgamatedproducts... ... Read more

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This book, first published in 1990, gives a general account of diagram manipulation techniques, as alternatives to algebraic methods of proof, in theoretical physics. Methods reviewed by the author include the popular techniques pioneered by Jucys and collaborators in the quantum theory of angular momentum and by Feynman in quantum field theory. The reader is encouraged to become bilingual in that many steps in the argument are presented as Problems, and are immediately followed by solutions and by comments on the method or proof and the significance of the results. This book will be of value to graduate students and research workers in theoretical solid state physics, atomic, molecular, nuclear and particle physics and theoretical chemistry. ... Read more

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A new approach to the character theory of the symmetric group has been developed during the past fifteen years which is in many ways more efficient, more transparent, and more elementary. In this approach, to each permutation is assigned a class function of the corresponding symmetric group. Problems in character theory can thereby be transferred into a completely different setting and reduced to combinatorial problems on permutations in a natural and uniform way.

This is the first account in book form entirely devoted to the new noncommutative method . As a modern and comprehensive survey of the classical theory the book contains such fundamental results as the Murnaghan Nakayama and Littlewood Richardson rules as well as more recent applications in enumerative combinatorics and in the theory of the free Lie algebra. But it is also an introduction to the vibrant theory of certain combinatorial Hopf algebras such as the Malvenuto Reutenauer algebra of permutations.

The three detailed appendices on group characters, the Solomon descent algebra and the Robinson Schensted correspondence makes the material self-contained and suitable for undergraduate level. Students and researchers alike will find that noncommutative character theory is a source of inspiration and an illuminating approach to this versatile field of algebraic combinatorics.

Contents: The Inductive Method; Noncommutative Character Theory of the Symmetric Group; Classical Character Theory of the Symmetric Group; Appendices: Elements of Representation Theory; Solomon's Mackey Formula; Young Tableaux and Knuth Relations. ... Read more

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This book brings together many of the important results in this field.

From the reviews: ""A classic gets even better....The edition has new material including the Novelli-Pak-Stoyanovskii bijective proof of the hook formula, Stanley’s proof of the sum of squares formula using differential posets, Fomin’s bijective proof of the sum of squares formula, group acting on posets and their use in proving unimodality, and chromatic symmetric functions." --ZENTRALBLATT MATH

Customer Reviews (4)

Worth the price just for the first chapter
Sagans book makes representation theory easy.The book first covers representations using modules and then choosing a basis to show the matrix approach.With every new topic he develops it using what Doron Zeilberger has dubbed the Gelfand Principle ([...])The principle is: "Always chooses the smallest example to make a point".It isn't easy to find the smallest example when Sn grows as quickly as it does, but Sagen always manages to do it.

The ensuing chapters follow in the same vein.Ideas are introduced and explained, sometimes with pictures, sometimes with calculations, but always as clearly as can be.

To read this book does require a firm grounding in linear algebra, as well as abstract algebra.Time reading it is time well spent.

Near Perfect
This book is excellent. The material is presented clearly and concisely. It makes the subject matter accessible and interesting. I used it as the text for a one-semester graduate subject. I completed all of the exercises, so it is well-paced for this kind of study. I started with only an introductory knowledge of group theory, so it is self-contained. The only drawback is that there are no solutions to any of the exercises. If it had this, it would be a perfect bok.

Good introduction for representation theory.
This book has 4 chapters.Chapter1 is about general theory of representations of finite group.Chapter2 is about representation of symmetric groups.chapter3 and 4 are about combinatorial topics and symmetric functions. Though I haven't read all of the book,I highlyrecommand this book because this book shows us introductive part ofrepresentation theory with easy words.I think it is worth to read for allwho are to begin the study of representation theory.

Good introduction for representation theory.
This book has 4 chapters.Chapter1 is about general theory of representations of finite group.Chapter2 is about representation of symmetric groups.chapter3 and 4 are about combinatorial topics and symmetric functions. Though I haven't read all of the book,I highlyrecommand this book because this book shows us introductive part ofrepresentation theory with easy words.I think it is worth to read for allwho are to begin the study of representation theory. ... Read more

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This new, more streamlined version of the 1999 third edition brings the existing materials and references up to date and omits information now readily available online and elsewhere.

The updated material in The Counselor and the Group makes this book an excellent resource for those who are both learning and practicing by providing a structured problem-solving approach to group work. Trotzer provides process and practice guidelines and techniques that enable group leaders to function effectively across the broad range of groups that counselors conduct including counseling, therapy, psychoeducational, and task groups. Includes material by noted group experts

• Lynn Rapin and Robert Conyne on "Best Practices in Group Counseling"
• Niloufer Merchant on Multicultural Counseling
• Rex Stockton, Paul Toth and D. Keith Morran on "The Case for Group Research."

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This classic in the field of abstract algebra has been called "the first systematic treatment of finite fields in the mathematical literature." Divided into two sections-"Introduction to the Galois Field Theory" and "Theory of Linear Groups in a Galois Field"-and crammed with examples and theorems, Linear Groups remains relevant more than a century after its publication. Learn about:. definition and properties of finite fields. classification and determination of irreducible quantics. miscellaneous properties of Galois Fields. analytic representation of substitutions on the marks of a Galois Field. the Abelian linear group. the hyperabelian group. the hyperorthogonal and related linear groups. and much more.This historic contribution to the theory of groups is still appreciated by mathematicians today.American mathematician LEONARD EUGENE DICKSON (1874-1954) received the University of Chicago's first doctorate in mathematics, and later taught there for 39 years. His published works include Linear Algebras (1914), History of the Theory of Numbers (1919-23), and Studies in the Theory of Numbers (1930). ... Read more

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From the reviews: "..., the book must be of great help for a researcher who already has some idea of Lie theory, wants to employ it in his everyday research and/or teaching, and needs a source for customary reference on the subject. From my viewpoint, the volume is perfectly fit to serve as such a source, ... On the whole, it is quite a pleasure, after making yourself comfortable in that favourite office armchair of yours, just to keep the volume gently in your hands and browse it slowly and thoughtfully; and after all, what more on Earth can one expect of any book?" The New Zealand Mathematical Society Newsletter

"... Both parts are very nicely written and can be strongly recommended." European Mathematical Society ... Read more

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Advanced textbook outlining combinatorial group theory, with emphasis on connections with topology, logic, and arguments of a primitive geometric nature. Also briefly mentions a number of topics not belonging to combinatorial group theory, such as groups acting on trees and linear groups. Softcover. ... Read more

Customer Reviews (3)

A reprint at last!
A text truly deserving the designation "classic"; bravo to Springer for reprinting this book!This seminal text provides a clear exposition of the topic, along with numerous jumping-off points for exploring interesting outgrowths and tangents.A must-have text for anyone interested in the subject.

Great book
A "must read" for everybody who is interested in combinatorial aspects of algebra. You can read whole book or some chapters (free groups, free constructions, diagrams, 1-relator, ...). I don't think it's forabsolute beginners, but if you want, you understand it. It would be greatif this book were reprinted with some complements (which could be writtenby Ol'shanskii, Gersten, Sapir,Shupp, Stallings, Guba, S.Ivanov,Baumslag, Lysyonok ...)

A very good textbook
It's an absolute pity that this book is out of print, since it's one of the few good recent textbooks about combinatorial group theory. One can only hope it will be reprinted soon... ... Read more