Finite Simple Groups Ii

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The Finite Simple Groups

Author : Robert Wilson
Publisher : Springer Science & Business Media
ISBN 13 : 1848009879
Total Pages : 310 pages
Book Rating : 4.8/5 (48 download)

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Book Synopsis The Finite Simple Groups by : Robert Wilson

Download or read book The Finite Simple Groups written by Robert Wilson and published by Springer Science & Business Media. This book was released on 2009-12-14 with total page 310 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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].

The Classification of the Finite Simple Groups, Number 3

Author : Daniel Gorenstein
Publisher : American Mathematical Soc.
ISBN 13 : 9780821803912
Total Pages : 446 pages
Book Rating : 4.8/5 (39 download)

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Book Synopsis The Classification of the Finite Simple Groups, Number 3 by : Daniel Gorenstein

Download or read book The Classification of the Finite Simple Groups, Number 3 written by Daniel Gorenstein and published by American Mathematical Soc.. This book was released on 1994 with total page 446 pages. Available in PDF, EPUB and Kindle. Book excerpt: Examines the internal structure of the finite simple groups of Lie type, the finite alternating groups, and 26 sporadic finite simple groups, as well as their analogues. Emphasis is on the structure of local subgroups and their relationships with one another, rather than development of an abstract theory of simple groups. A foundation is laid for the development of specific properties of K-groups to be used in the inductive proof of the classification theorem. Highlights include statements and proofs of the Breol-Tits and Curtis-Tits theorems, and material on centralizers of semisimple involutions in groups of Lie type. For graduate students and research mathematicians. Annotation copyrighted by Book News, Inc., Portland, OR

The Classification of Finite Simple Groups

Author : Michael Aschbacher
Publisher : American Mathematical Soc.
ISBN 13 : 0821853368
Total Pages : 362 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis The Classification of Finite Simple Groups by : Michael Aschbacher

Download or read book The Classification of Finite Simple Groups written by Michael Aschbacher and published by American Mathematical Soc.. This book was released on 2011 with total page 362 pages. Available in PDF, EPUB and Kindle. Book excerpt: Provides an outline and modern overview of the classification of the finite simple groups. It primarily covers the 'even case', where the main groups arising are Lie-type (matrix) groups over a field of characteristic 2. The book thus completes a project begun by Daniel Gorenstein's 1983 book, which outlined the classification of groups of 'noncharacteristic 2 type'.

Finite Simple Groups

Author : Daniel Gorenstein
Publisher : Springer Science & Business Media
ISBN 13 : 1468484974
Total Pages : 339 pages
Book Rating : 4.4/5 (684 download)

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Book Synopsis Finite Simple Groups by : Daniel Gorenstein

Download or read book Finite Simple Groups written by Daniel Gorenstein and published by Springer Science & Business Media. This book was released on 2013-11-27 with total page 339 pages. Available in PDF, EPUB and Kindle. Book excerpt: In February 1981, the classification of the finite simple groups (Dl)* was completed,t. * representing one of the most remarkable achievements in the history or mathematics. Involving the combined efforts of several hundred mathematicians from around the world over a period of 30 years, the full proof covered something between 5,000 and 10,000 journal pages, spread over 300 to 500 individual papers. The single result that, more than any other, opened up the field and foreshadowed the vastness of the full classification proof was the celebrated theorem of Walter Feit and John Thompson in 1962, which stated that every finite group of odd order (D2) is solvable (D3)-a statement expressi ble in a single line, yet its proof required a full 255-page issue of the Pacific 10urnal of Mathematics [93]. Soon thereafter, in 1965, came the first new sporadic simple group in over 100 years, the Zvonimir Janko group 1 , to further stimulate the 1 'To make the book as self-contained as possible. we are including definitions of various terms as they occur in the text. However. in order not to disrupt the continuity of the discussion. we have placed them at the end of the Introduction. We denote these definitions by (DI). (D2), (D3). etc.

Finite Simple Groups: Thirty Years of the Atlas and Beyond

Author : Manjul Bhargava
Publisher : American Mathematical Soc.
ISBN 13 : 1470436787
Total Pages : 242 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Finite Simple Groups: Thirty Years of the Atlas and Beyond by : Manjul Bhargava

Download or read book Finite Simple Groups: Thirty Years of the Atlas and Beyond written by Manjul Bhargava and published by American Mathematical Soc.. This book was released on 2017-07-24 with total page 242 pages. Available in PDF, EPUB and Kindle. Book excerpt: Classification of Finite Simple Groups, one of the most monumental accomplishments of modern mathematics, was announced in 1983 with the proof completed in 2004. Since then, it has opened up a new and powerful strategy to approach and resolve many previously inaccessible problems in group theory, number theory, combinatorics, coding theory, algebraic geometry, and other areas of mathematics. This strategy crucially utilizes various information about finite simple groups, part of which is catalogued in the Atlas of Finite Groups (John H. Conway et al.), and in An Atlas of Brauer Characters (Christoph Jansen et al.). It is impossible to overestimate the roles of the Atlases and the related computer algebra systems in the everyday life of researchers in many areas of contemporary mathematics. The main objective of the conference was to discuss numerous applications of the Atlases and to explore recent developments and future directions of research, with focus on the interaction between computation and theory and applications to number theory and algebraic geometry. The papers in this volume are based on talks given at the conference. They present a comprehensive survey on current research in all of these fields.

Theory of Finite Simple Groups

Author : Gerhard Michler
Publisher : Cambridge University Press
ISBN 13 : 0521866251
Total Pages : 638 pages
Book Rating : 4.5/5 (218 download)

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Book Synopsis Theory of Finite Simple Groups by : Gerhard Michler

Download or read book Theory of Finite Simple Groups written by Gerhard Michler and published by Cambridge University Press. This book was released on 2006-09-21 with total page 638 pages. Available in PDF, EPUB and Kindle. Book excerpt: The first representation theoretic and algorithmic approach to the theory of abstract finite simple groups.

The Subgroup Structure of the Finite Classical Groups

Author : Peter B. Kleidman
Publisher : Cambridge University Press
ISBN 13 : 052135949X
Total Pages : 317 pages
Book Rating : 4.5/5 (213 download)

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Book Synopsis The Subgroup Structure of the Finite Classical Groups by : Peter B. Kleidman

Download or read book The Subgroup Structure of the Finite Classical Groups written by Peter B. Kleidman and published by Cambridge University Press. This book was released on 1990-04-26 with total page 317 pages. Available in PDF, EPUB and Kindle. Book excerpt: With the classification of the finite simple groups complete, much work has gone into the study of maximal subgroups of almost simple groups. In this volume the authors investigate the maximal subgroups of the finite classical groups and present research into these groups as well as proving many new results. In particular, the authors develop a unified treatment of the theory of the 'geometric subgroups' of the classical groups, introduced by Aschbacher, and they answer the questions of maximality and conjugacy and obtain the precise shapes of these groups. Both authors are experts in the field and the book will be of considerable value not only to group theorists, but also to combinatorialists and geometers interested in these techniques and results. Graduate students will find it a very readable introduction to the topic and it will bring them to the very forefront of research in group theory.

The Classification of Finite Simple Groups

Author : Daniel Gorenstein
Publisher : Springer Science & Business Media
ISBN 13 : 1461336856
Total Pages : 493 pages
Book Rating : 4.4/5 (613 download)

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Book Synopsis The Classification of Finite Simple Groups by : Daniel Gorenstein

Download or read book The Classification of Finite Simple Groups written by Daniel Gorenstein and published by Springer Science & Business Media. This book was released on 2013-11-22 with total page 493 pages. Available in PDF, EPUB and Kindle. Book excerpt: Never before in the history of mathematics has there been an individual theorem whose proof has required 10,000 journal pages of closely reasoned argument. Who could read such a proof, let alone communicate it to others? But the classification of all finite simple groups is such a theorem-its complete proof, developed over a 30-year period by about 100 group theorists, is the union of some 500 journal articles covering approximately 10,000 printed pages. How then is one who has lived through it all to convey the richness and variety of this monumental achievement? Yet such an attempt must be made, for without the existence of a coherent exposition of the total proof, there is a very real danger that it will gradually become lost to the living world of mathematics, buried within the dusty pages of forgotten journals. For it is almost impossible for the uninitiated to find the way through the tangled proof without an experienced guide; even the 500 papers themselves require careful selection from among some 2,000 articles on simple group theory, which together include often attractive byways, but which serve only to delay the journey.

Symmetric Generation of Groups

Author : Robert Curtis
Publisher : Cambridge University Press
ISBN 13 : 052185721X
Total Pages : 333 pages
Book Rating : 4.5/5 (218 download)

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Book Synopsis Symmetric Generation of Groups by : Robert Curtis

Download or read book Symmetric Generation of Groups written by Robert Curtis and published by Cambridge University Press. This book was released on 2007-07-05 with total page 333 pages. Available in PDF, EPUB and Kindle. Book excerpt: Comprehensive text which develops the notion of symmetric generation and applies the technique to sporadic simple groups.

Finite Groups

Author : Daniel Gorenstein
Publisher : American Mathematical Soc.
ISBN 13 : 0821843427
Total Pages : 546 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Finite Groups by : Daniel Gorenstein

Download or read book Finite Groups written by Daniel Gorenstein and published by American Mathematical Soc.. This book was released on 2007 with total page 546 pages. Available in PDF, EPUB and Kindle. Book excerpt: "The Classification Theorem is one of the main achievements of 20th century mathematics, but its proof has not yet been completely extricated from the journal literature in which it first appeared. This is the second volume in a series devoted to the presentation of a reorganized and simplified proof of the classification of the finite simple groups. The authors present (with either proof or reference to a proof) those theorems of abstract finite group theory, which are fundamental to the analysis in later volumes in the series. This volume provides a relatively concise and readable access to the key ideas and theorems underlying the study of finite simple groups and their important subgroups. The sections on semisimple subgroups and subgroups of parabolic type give detailed treatments of these important subgroups, including some results not available until now or available only in journal literature. The signalizer section provides an extensive development of both the Bender Method and the Signalizer Functor Method, which play a central role in the proof of the Classification Theorem. This book would be a valuable companion text for a graduate group theory course."--Publisher's website

Finite Group Theory

Author : M. Aschbacher
Publisher : Cambridge University Press
ISBN 13 : 9780521786751
Total Pages : 320 pages
Book Rating : 4.7/5 (867 download)

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Book Synopsis Finite Group Theory by : M. Aschbacher

Download or read book Finite Group Theory written by M. Aschbacher and published by Cambridge University Press. This book was released on 2000-06-26 with total page 320 pages. Available in PDF, EPUB and Kindle. Book excerpt: During the last 40 years the theory of finite groups has developed dramatically. The finite simple groups have been classified and are becoming better understood. Tools exist to reduce many questions about arbitrary finite groups to similar questions about simple groups. Since the classification there have been numerous applications of this theory in other branches of mathematics. Finite Group Theory develops the foundations of the theory of finite groups. It can serve as a text for a course on finite groups for students already exposed to a first course in algebra. It could supply the background necessary to begin reading journal articles in the field. For specialists it also provides a reference on the foundations of the subject. 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.

Applications of Finite Groups

Author : J. S. Lomont
Publisher : Academic Press
ISBN 13 : 1483268969
Total Pages : 359 pages
Book Rating : 4.4/5 (832 download)

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Book Synopsis Applications of Finite Groups by : J. S. Lomont

Download or read book Applications of Finite Groups written by J. S. Lomont and published by Academic Press. This book was released on 2014-05-12 with total page 359 pages. Available in PDF, EPUB and Kindle. Book excerpt: Applications of Finite Groups focuses on the applications of finite groups to problems of physics, including representation theory, crystals, wave equations, and nuclear and molecular structures. The book first elaborates on matrices, groups, and representations. Topics include abstract properties, applications, matrix groups, key theorem of representation theory, properties of character tables, simply reducible groups, tensors and invariants, and representations generated by functions. The text then examines applications and subgroups and representations, as well as subduced and induced representations, fermion annihilation and creation operators, crystallographic point groups, proportionality tensors in crystals, and nonrelativistic wave equations. The publication takes a look at space group representations and energy bands, symmetric groups, and applications. Topics include molecular and nuclear structures, multiplet splitting in crystalline electric fields, construction of irreducible representations of the symmetric groups, and reality of representations. The manuscript is a dependable source of data for physicists and researchers interested in the applications of finite groups.

Generators and Relations in Groups and Geometries

Author : A. Barlotti
Publisher : Springer Science & Business Media
ISBN 13 : 9401133824
Total Pages : 455 pages
Book Rating : 4.4/5 (11 download)

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Book Synopsis Generators and Relations in Groups and Geometries by : A. Barlotti

Download or read book Generators and Relations in Groups and Geometries written by A. Barlotti and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 455 pages. Available in PDF, EPUB and Kindle. Book excerpt: Every group is represented in many ways as an epimorphic image of a free group. It seems therefore futile to search for methods involving generators and relations which can be used to detect the structure of a group. Nevertheless, results in the indicated direction exist. The clue is to ask the right question. Classical geometry is a typical example in which the factorization of a motion into reflections or, more generally, of a collineation into central collineations, supplies valuable information on the geometric and algebraic structure. This mode of investigation has gained momentum since the end of last century. The tradition of geometric-algebraic interplay brought forward two branches of research which are documented in Parts I and II of these Proceedings. Part II deals with the theory of reflection geometry which culminated in Bachmann's work where the geometric information is encoded in properties of the group of motions expressed by relations in the generating involutions. This approach is the backbone of the classification of motion groups for the classical unitary and orthogonal planes. The axioms in this char acterization are natural and plausible. They provoke the study of consequences of subsets of axioms which also yield natural geometries whose exploration is rewarding. Bachmann's central axiom is the three reflection theorem, showing that the number of reflections needed to express a motion is of great importance.

The Classification of the Finite Simple Groups, Number 9

Author : Inna Capdeboscq
Publisher : American Mathematical Society
ISBN 13 : 1470464373
Total Pages : 520 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis The Classification of the Finite Simple Groups, Number 9 by : Inna Capdeboscq

Download or read book The Classification of the Finite Simple Groups, Number 9 written by Inna Capdeboscq and published by American Mathematical Society. This book was released on 2021-02-22 with total page 520 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is the ninth volume in a series whose goal is to furnish a careful and largely self-contained proof of the classification theorem for the finite simple groups. Having completed the classification of the simple groups of odd type as well as the classification of the simple groups of generic even type (modulo uniqueness theorems to appear later), the current volume begins the classification of the finite simple groups of special even type. The principal result of this volume is a classification of the groups of bicharacteristic type, i.e., of both even type and of $p$-type for a suitable odd prime $p$. It is here that the largest sporadic groups emerge, namely the Monster, the Baby Monster, the largest Conway group, and the three Fischer groups, along with six finite groups of Lie type over small fields, several of which play a major role as subgroups or sections of these sporadic groups.

Atlas of Finite Groups

Author : John Horton Conway
Publisher : Oxford University Press
ISBN 13 : 9780198531999
Total Pages : 252 pages
Book Rating : 4.5/5 (319 download)

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Book Synopsis Atlas of Finite Groups by : John Horton Conway

Download or read book Atlas of Finite Groups written by John Horton Conway and published by Oxford University Press. This book was released on 1985 with total page 252 pages. Available in PDF, EPUB and Kindle. Book excerpt: This atlas covers groups from the families of the classification of finite simple groups. Recently updated incorporating corrections

A Course in Finite Group Representation Theory

Author : Peter Webb
Publisher : Cambridge University Press
ISBN 13 : 1107162394
Total Pages : 339 pages
Book Rating : 4.1/5 (71 download)

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Book Synopsis A Course in Finite Group Representation Theory by : Peter Webb

Download or read book A Course in Finite Group Representation Theory written by Peter Webb and published by Cambridge University Press. This book was released on 2016-08-19 with total page 339 pages. Available in PDF, EPUB and Kindle. Book excerpt: This graduate-level text provides a thorough grounding in the representation theory of finite groups over fields and rings. The book provides a balanced and comprehensive account of the subject, detailing the methods needed to analyze representations that arise in many areas of mathematics. Key topics include the construction and use of character tables, the role of induction and restriction, projective and simple modules for group algebras, indecomposable representations, Brauer characters, and block theory. This classroom-tested text provides motivation through a large number of worked examples, with exercises at the end of each chapter that test the reader's knowledge, provide further examples and practice, and include results not proven in the text. Prerequisites include a graduate course in abstract algebra, and familiarity with the properties of groups, rings, field extensions, and linear algebra.

Modular Representation Theory of Finite Groups

Author : Peter Schneider
Publisher : Springer Science & Business Media
ISBN 13 : 1447148320
Total Pages : 183 pages
Book Rating : 4.4/5 (471 download)

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Book Synopsis Modular Representation Theory of Finite Groups by : Peter Schneider

Download or read book Modular Representation Theory of Finite Groups written by Peter Schneider and published by Springer Science & Business Media. This book was released on 2012-11-27 with total page 183 pages. Available in PDF, EPUB and Kindle. Book excerpt: Representation theory studies maps from groups into the general linear group of a finite-dimensional vector space. For finite groups the theory comes in two distinct flavours. In the 'semisimple case' (for example over the field of complex numbers) one can use character theory to completely understand the representations. This by far is not sufficient when the characteristic of the field divides the order of the group. Modular Representation Theory of finite Groups comprises this second situation. Many additional tools are needed for this case. To mention some, there is the systematic use of Grothendieck groups leading to the Cartan matrix and the decomposition matrix of the group as well as Green's direct analysis of indecomposable representations. There is also the strategy of writing the category of all representations as the direct product of certain subcategories, the so-called 'blocks' of the group. Brauer's work then establishes correspondences between the blocks of the original group and blocks of certain subgroups the philosophy being that one is thereby reduced to a simpler situation. In particular, one can measure how nonsemisimple a category a block is by the size and structure of its so-called 'defect group'. All these concepts are made explicit for the example of the special linear group of two-by-two matrices over a finite prime field. Although the presentation is strongly biased towards the module theoretic point of view an attempt is made to strike a certain balance by also showing the reader the group theoretic approach. In particular, in the case of defect groups a detailed proof of the equivalence of the two approaches is given. This book aims to familiarize students at the masters level with the basic results, tools, and techniques of a beautiful and important algebraic theory. Some basic algebra together with the semisimple case are assumed to be known, although all facts to be used are restated (without proofs) in the text. Otherwise the book is entirely self-contained.