An Introduction to the Theory of Groups

An Introduction to the Theory of Groups

Anyone who has studied abstract algebra and linear algebra as an undergraduate can understand this book. The first six chapters provide material for a first course, while the rest of the book covers more advanced topics.

Author: Joseph J. Rotman

Publisher: Springer Science & Business Media

ISBN: 9781461241768

Category: Mathematics

Page: 517

View: 160

Anyone who has studied abstract algebra and linear algebra as an undergraduate can understand this book. The first six chapters provide material for a first course, while the rest of the book covers more advanced topics. This revised edition retains the clarity of presentation that was the hallmark of the previous editions. From the reviews: "Rotman has given us a very readable and valuable text, and has shown us many beautiful vistas along his chosen route." --MATHEMATICAL REVIEWS
Categories: Mathematics

A Course in the Theory of Groups

A Course in the Theory of Groups

"An excellent up-to-date introduction to the theory of groups.

Author: Derek J.S. Robinson

Publisher: Springer Science & Business Media

ISBN: 9781441985941

Category: Mathematics

Page: 502

View: 572

"An excellent up-to-date introduction to the theory of groups. It is general yet comprehensive, covering various branches of group theory. The 15 chapters contain the following main topics: free groups and presentations, free products, decompositions, Abelian groups, finite permutation groups, representations of groups, finite and infinite soluble groups, group extensions, generalizations of nilpotent and soluble groups, finiteness properties." —-ACTA SCIENTIARUM MATHEMATICARUM
Categories: Mathematics

Introduction to Group Theory

Introduction to Group Theory

This book quickly introduces beginners to general group theory and then focuses on three main themes : finite group theory, including sporadic groups combinatorial and geometric group theory, including the Bass-Serre theory of groups acting ...

Author: Oleg Vladimirovič Bogopolʹskij

Publisher: European Mathematical Society

ISBN: 3037190418

Category: Combinatorial group theory

Page: 177

View: 604

This book quickly introduces beginners to general group theory and then focuses on three main themes : finite group theory, including sporadic groups combinatorial and geometric group theory, including the Bass-Serre theory of groups acting on trees the theory of train tracks by Bestvina and Handel for automorphisms of free groups With its many examples, exercises, and full solutions to selected exercises, this text provides a gentle introduction that is ideal for self-study and an excellent preparation for applications. A distinguished feature of the presentation is that algebraic and geometric techniques are balanced. The beautiful theory of train tracks is illustrated by two nontrivial examples. Presupposing only a basic knowledge of algebra, the book is addressed to anyone interested in group theory: from advanced undergraduate and graduate students to specialists.
Categories: Combinatorial group theory

Model Theory An Introduction

Model Theory   An Introduction

Assumes only a familiarity with algebra at the beginning graduate level; Stresses applications to algebra; Illustrates several of the ways Model Theory can be a useful tool in analyzing classical mathematical structures This book is a ...

Author: David Marker

Publisher: Springer Science & Business Media

ISBN: 9780387227344

Category: Mathematics

Page: 345

View: 510

Assumes only a familiarity with algebra at the beginning graduate level; Stresses applications to algebra; Illustrates several of the ways Model Theory can be a useful tool in analyzing classical mathematical structures
Categories: Mathematics

An Introduction to Knot Theory

An Introduction to Knot Theory

Readers are assumed to have knowledge of the basic ideas of the fundamental group and simple homology theory, although explanations throughout the text are numerous and well-done.

Author: W.B.Raymond Lickorish

Publisher: Springer Science & Business Media

ISBN: 9781461206910

Category: Mathematics

Page: 204

View: 133

A selection of topics which graduate students have found to be a successful introduction to the field, employing three distinct techniques: geometric topology manoeuvres, combinatorics, and algebraic topology. Each topic is developed until significant results are achieved and each chapter ends with exercises and brief accounts of the latest research. What may reasonably be referred to as knot theory has expanded enormously over the last decade and, while the author describes important discoveries throughout the twentieth century, the latest discoveries such as quantum invariants of 3-manifolds as well as generalisations and applications of the Jones polynomial are also included, presented in an easily intelligible style. Readers are assumed to have knowledge of the basic ideas of the fundamental group and simple homology theory, although explanations throughout the text are numerous and well-done. Written by an internationally known expert in the field, this will appeal to graduate students, mathematicians and physicists with a mathematical background wishing to gain new insights in this area.
Categories: Mathematics

Topological Methods in Group Theory

Topological Methods in Group Theory

This book is about the interplay between algebraic topology and the theory of infinite discrete groups.

Author: Ross Geoghegan

Publisher: Springer Science & Business Media

ISBN: 9780387746111

Category: Mathematics

Page: 473

View: 788

This book is about the interplay between algebraic topology and the theory of infinite discrete groups. It is a hugely important contribution to the field of topological and geometric group theory, and is bound to become a standard reference in the field. To keep the length reasonable and the focus clear, the author assumes the reader knows or can easily learn the necessary algebra, but wants to see the topology done in detail. The central subject of the book is the theory of ends. Here the author adopts a new algebraic approach which is geometric in spirit.
Categories: Mathematics

Lie Groups

Lie Groups

Lie Groups is intended as an introduction to the theory of Lie groups and their representations at the advanced undergraduate or beginning graduate level.

Author: Wulf Rossmann

Publisher: Oxford University Press on Demand

ISBN: 0199202516

Category: Mathematics

Page: 265

View: 223

Lie Groups is intended as an introduction to the theory of Lie groups and their representations at the advanced undergraduate or beginning graduate level. It covers the essentials of the subject starting from basic undergraduate mathematics. The correspondence between linear Lie groups andLie algebras is developed in its local and global aspects. The classical groups are analysed in detail, first with elementary matrix methods, then with the help of the structural tools typical of the theory of semisimple groups, such as Cartan subgroups, roots, weights, and reflections. Thefundamental groups of the classical groups are worked out as an application of these methods. Manifolds are introduced when needed, in connection with homogeneous spaces, and the elements of differential and integral calculus on manifolds are presented, with special emphasis on integration ongroups and homogeneous spaces. Representation theory starts from first principles, such as Schur's lemma and its consequences, and proceeds from there to the Peter-Weyl theorem, Weyl's character formula, and the Borel-Weil theorem, all in the context of linear groups.
Categories: Mathematics

A Course in Group Theory

A Course in Group Theory

Each chapter ends with a summary of the material covered and notes on the history and development of group theory.

Author: J. F. Humphreys

Publisher: Oxford University Press on Demand

ISBN: 0198534590

Category: Business & Economics

Page: 279

View: 265

Each chapter ends with a summary of the material covered and notes on the history and development of group theory.
Categories: Business & Economics

An Introduction to Quantum Theory

An Introduction to Quantum Theory

This book is intended for the later years of an undergraduate course or for graduates. It assumes a knowledge of basic linear algebra and elementary group theory, though for convenience these are also summarized in an appendix.

Author: Keith Hannabuss

Publisher: Clarendon Press

ISBN: 0191588733

Category: Science

Page: 394

View: 419

This book provides an introduction to quantum theory primarily for students of mathematics. Although the approach is mainly traditional the discussion exploits ideas of linear algebra, and points out some of the mathematical subtleties of the theory. Amongst the less traditional topics are Bell's inequalities, coherent and squeezed states, and introductions to group representation theory. Later chapters discuss relativistic wave equations and elementary particle symmetries from a group theoretical standpoint rather than the customary Lie algebraic approach. This book is intended for the later years of an undergraduate course or for graduates. It assumes a knowledge of basic linear algebra and elementary group theory, though for convenience these are also summarized in an appendix.
Categories: Science

Linear Algebraic Groups

Linear Algebraic Groups

His main research interests include group theory and Lie algebras, and this graduate level text is an exceptionally well-written introduction to everything about linear algebraic groups.

Author: James E. Humphreys

Publisher: Springer Science & Business Media

ISBN: 9781468494433

Category: Mathematics

Page: 248

View: 848

James E. Humphreys is a distinguished Professor of Mathematics at the University of Massachusetts at Amherst. He has previously held posts at the University of Oregon and New York University. His main research interests include group theory and Lie algebras, and this graduate level text is an exceptionally well-written introduction to everything about linear algebraic groups.
Categories: Mathematics

Cohomology of Groups

Cohomology of Groups

Aimed at second year graduate students, this text introduces them to cohomology theory (involving a rich interplay between algebra and topology) with a minimum of prerequisites.

Author: Kenneth S. Brown

Publisher: Springer Science & Business Media

ISBN: 9781468493276

Category: Mathematics

Page: 306

View: 303

Aimed at second year graduate students, this text introduces them to cohomology theory (involving a rich interplay between algebra and topology) with a minimum of prerequisites. No homological algebra is assumed beyond what is normally learned in a first course in algebraic topology, and the basics of the subject, as well as exercises, are given prior to discussion of more specialized topics.
Categories: Mathematics

A Basic Course in Algebraic Topology

A Basic Course in Algebraic Topology

The text consists of material from the first five chapters of the author's earlier book, Algebraic Topology; an Introduction (GTM 56) together with almost all of his book, Singular Homology Theory (GTM 70).

Author: William S. Massey

Publisher: Springer

ISBN: 9781493990634

Category: Mathematics

Page: 431

View: 430

This textbook is intended for a course in algebraic topology at the beginning graduate level. The main topics covered are the classification of compact 2-manifolds, the fundamental group, covering spaces, singular homology theory, and singular cohomology theory. These topics are developed systematically, avoiding all unnecessary definitions, terminology, and technical machinery. The text consists of material from the first five chapters of the author's earlier book, Algebraic Topology; an Introduction (GTM 56) together with almost all of his book, Singular Homology Theory (GTM 70). The material from the two earlier books has been substantially revised, corrected, and brought up to date.
Categories: Mathematics

Topics in Geometric Group Theory

Topics in Geometric Group Theory

S. Rosset, A property of groups of nonerponential growth, Proc. Amer. Math. Soc.
54 (1976), 24–26 {VII.27}. J.J. Rotman, An introduction to the theory of groups,
fourth edition, Graduate Texts in Mathematics 148, Springer, 1995 (first edition ...

Author: Pierre de la Harpe

Publisher: University of Chicago Press

ISBN: 0226317196

Category: Mathematics

Page: 310

View: 182

In this book, Pierre de la Harpe provides a concise and engaging introduction to geometric group theory, a new method for studying infinite groups via their intrinsic geometry that has played a major role in mathematics over the past two decades. A recognized expert in the field, de la Harpe adopts a hands-on approach, illustrating key concepts with numerous concrete examples. The first five chapters present basic combinatorial and geometric group theory in a unique and refreshing way, with an emphasis on finitely generated versus finitely presented groups. In the final three chapters, de la Harpe discusses new material on the growth of groups, including a detailed treatment of the "Grigorchuk group." Most sections are followed by exercises and a list of problems and complements, enhancing the book's value for students; problems range from slightly more difficult exercises to open research problems in the field. An extensive list of references directs readers to more advanced results as well as connections with other fields.
Categories: Mathematics

Introduction to Lie Algebras and Representation Theory

Introduction to Lie Algebras and Representation Theory

This book is designed to introduce the reader to the theory of semisimple Lie algebras over an algebraically closed field of characteristic 0, with emphasis on representations.

Author: JAMES HUMPHREYS

Publisher: Springer Science & Business Media

ISBN: 0387900535

Category: Mathematics

Page: 173

View: 489

This book is designed to introduce the reader to the theory of semisimple Lie algebras over an algebraically closed field of characteristic 0, with emphasis on representations. A good knowledge of linear algebra (including eigenvalues, bilinear forms, euclidean spaces, and tensor products of vector spaces) is presupposed, as well as some acquaintance with the methods of abstract algebra. The first four chapters might well be read by a bright undergraduate; however, the remaining three chapters are admittedly a little more demanding. Besides being useful in many parts of mathematics and physics, the theory of semisimple Lie algebras is inherently attractive, combining as it does a certain amount of depth and a satisfying degree of completeness in its basic results. Since Jacobson's book appeared a decade ago, improvements have been made even in the classical parts of the theory. I have tried to incor porate some of them here and to provide easier access to the subject for non-specialists. For the specialist, the following features should be noted: (I) The Jordan-Chevalley decomposition of linear transformations is emphasized, with "toral" subalgebras replacing the more traditional Cartan subalgebras in the semisimple case. (2) The conjugacy theorem for Cartan subalgebras is proved (following D. J. Winter and G. D. Mostow) by elementary Lie algebra methods, avoiding the use of algebraic geometry.
Categories: Mathematics

An Introduction to the Representation Theory of Groups

An Introduction to the Representation Theory of Groups

The goal of this book is to present, in a motivated manner, the basic formalism of representation theory as well as some important applications.

Author: Emmanuel Kowalski

Publisher: American Mathematical Society

ISBN: 9781470409661

Category: Mathematics

Page: 432

View: 836

Representation theory is an important part of modern mathematics, not only as a subject in its own right but also as a tool for many applications. It provides a means for exploiting symmetry, making it particularly useful in number theory, algebraic geometry, and differential geometry, as well as classical and modern physics. The goal of this book is to present, in a motivated manner, the basic formalism of representation theory as well as some important applications. The style is intended to allow the reader to gain access to the insights and ideas of representation theory--not only to verify that a certain result is true, but also to explain why it is important and why the proof is natural. The presentation emphasizes the fact that the ideas of representation theory appear, sometimes in slightly different ways, in many contexts. Thus the book discusses in some detail the fundamental notions of representation theory for arbitrary groups. It then considers the special case of complex representations of finite groups and discusses the representations of compact groups, in both cases with some important applications. There is a short introduction to algebraic groups as well as an introduction to unitary representations of some noncompact groups. The text includes many exercises and examples.
Categories: Mathematics

Classical Topology and Combinatorial Group Theory

Classical Topology and Combinatorial Group Theory

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.

Author:

Publisher: Springer Science & Business Media

ISBN: 9781468401103

Category: Mathematics

Page: 301

View: 885

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.
Categories: Mathematics

The Finite Simple Groups

The Finite Simple Groups

This is the first time that all the finite simple groups have been treated together in this way and the book points out their connections, for example between exceptional behaviour of generic groups and the existence of sporadic groups, and ...

Author: Robert Wilson

Publisher: Springer Science & Business Media

ISBN: 9781848009875

Category: Mathematics

Page: 298

View: 126

Thisbookisintendedasanintroductiontoallthe?nitesimplegroups.During themonumentalstruggletoclassifythe?nitesimplegroups(andindeedsince), a huge amount of information about these groups has been accumulated. Conveyingthisinformationtothenextgenerationofstudentsandresearchers, not to mention those who might wish to apply this knowledge, has become a major challenge. With the publication of the two volumes by Aschbacher and Smith [12, 13] in 2004 we can reasonably regard the proof of the Classi?cation Theorem for Finite Simple Groups (usually abbreviated CFSG) as complete. Thus it is timely to attempt an overview of all the (non-abelian) ?nite simple groups in one volume. For expository purposes it is convenient to divide them into four basic types, namely the alternating, classical, exceptional and sporadic groups. The study of alternating groups soon develops into the theory of per- tation groups, which is well served by the classic text of Wielandt [170]and more modern treatments such as the comprehensive introduction by Dixon and Mortimer [53] and more specialised texts such as that of Cameron [19].
Categories: Mathematics

The Theory of Groups

The Theory of Groups

Many of the proofs in the text are shorter and more transparent than the usual, older ones, and a series of helpful appendixes presents material new to this edition.

Author: Hans Zassenhaus

Publisher: Courier Dover Publications

ISBN: 0486409228

Category: Mathematics

Page: 265

View: 251

Group theory represents one of the most fundamental elements of mathematics. Indispensable in nearly every branch of the field, concepts from the theory of groups also have important applications beyond mathematics, in such areas as quantum mechanics and crystallography. Hans J. Zassenhaus, a pioneer in the study of group theory, has designed this useful, well-written, graduate-level text to acquaint the reader with group-theoretic methods and to demonstrate their usefulness as tools in the solution of mathematical and physical problems. Starting with an exposition of the fundamental concepts of group theory, including an investigation of axioms, the calculus of complexes, and a theorem of Frobenius, the author moves on to a detailed investigation of the concept of homomorphic mapping, along with an examination of the structure and construction of composite groups from simple components. The elements of the theory of p-groups receive a coherent treatment, and the volume concludes with an explanation of a method by which solvable factor groups may be split off from a finite group. Many of the proofs in the text are shorter and more transparent than the usual, older ones, and a series of helpful appendixes presents material new to this edition. This material includes an account of the connections between lattice theory and group theory, and many advanced exercises illustrating both lattice-theoretical ideas and the extension of group-theoretical concepts to multiplicative domains.
Categories: Mathematics

Lie Groups Lie Algebras and Representations

Lie Groups  Lie Algebras  and Representations

This book addresses Lie groups, Lie algebras, and representation theory.

Author: Brian Hall

Publisher: Springer Science & Business Media

ISBN: 0387401229

Category: Mathematics

Page: 351

View: 145

This book addresses Lie groups, Lie algebras, and representation theory. The author restricts attention to matrix Lie groups and Lie algebras. This approach keeps the discussion concrete, allows the reader to get to the heart of the subject quickly, and covers all of the most interesting examples.From the reviews:"Sure to become a standard textbook for graduate students in mathematics and physics with little or no prior exposure to Lie theory." --L'Enseignement Mathematique
Categories: Mathematics

An Introduction to Algebraic Topology

An Introduction to Algebraic Topology

Although categories and functors are introduced early in the text, excessive generality is avoided, and the author explains the geometric or analytic origins of abstract concepts as they are introduced.

Author: Joseph Rotman

Publisher: Springer Science & Business Media

ISBN: 0387966781

Category: Mathematics

Page: 438

View: 344

A clear exposition, with exercises, of the basic ideas of algebraic topology. Suitable for a two-semester course at the beginning graduate level, it assumes a knowledge of point set topology and basic algebra. Although categories and functors are introduced early in the text, excessive generality is avoided, and the author explains the geometric or analytic origins of abstract concepts as they are introduced.
Categories: Mathematics