**Author**: Robert Wilson

**Publisher:** Springer Science & Business Media

**ISBN:** 1848009879

**Category : **Mathematics

**Languages : **en

**Pages : **298

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**Book Description**
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].

**Author**: Robert Wilson

**Publisher:** Springer Science & Business Media

**ISBN:** 1848009879

**Category : **Mathematics

**Languages : **en

**Pages : **298

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**Book Description**
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].

**Author**: Gerhard Michler

**Publisher:** Cambridge University Press

**ISBN:** 0521866251

**Category : **Mathematics

**Languages : **en

**Pages : **662

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**Book Description**
The first representation theoretic and algorithmic approach to the theory of abstract finite simple groups.

**Author**: Daniel Gorenstein

**Publisher:** American Mathematical Soc.

**ISBN:** 0821827774

**Category : **Mathematics

**Languages : **en

**Pages : **529

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**Book Description**
The classification of finite simple groups is a landmark result of modern mathematics. The original proof is spread over scores of articles by dozens of researchers. In this multivolume book, the authors have succeeded in the monumental task of assembling this original proof with explanations and references. The present book, along with background from sections of the previous volumes, presents critical aspects of the classification.

**Author**: Daniel Gorenstein

**Publisher:** American Mathematical Soc.

**ISBN:** 0821809601

**Category : **Mathematics

**Languages : **en

**Pages : **165

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**Book Description**

**Author**: Daniel Gorenstein

**Publisher:** American Mathematical Soc.

**ISBN:** 1470441896

**Category : **
**Languages : **en

**Pages : **488

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**Book Description**
This book completes a trilogy (Numbers 5, 7, and 8) of the series The Classification of the Finite Simple Groups treating the generic case of the classification of the finite simple groups. In conjunction with Numbers 4 and 6, it allows us to reach a major milestone in our series—the completion of the proof of the following theorem: Theorem O: Let G be a finite simple group of odd type, all of whose proper simple sections are known simple groups. Then either G is an alternating group or G is a finite group of Lie type defined over a field of odd order or G is one of six sporadic simple groups. Put another way, Theorem O asserts that any minimal counterexample to the classification of the finite simple groups must be of even type. The work of Aschbacher and Smith shows that a minimal counterexample is not of quasithin even type, while this volume shows that a minimal counterexample cannot be of generic even type, modulo the treatment of certain intermediate configurations of even type which will be ruled out in the next volume of our series.

**Author**: Inna Capdeboscq

**Publisher:** American Mathematical Society

**ISBN:** 1470464373

**Category : **Mathematics

**Languages : **en

**Pages : **520

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**Book Description**
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.

**Author**: Daniel Gorenstein

**Publisher:** American Mathematical Soc.

**ISBN:** 082181379X

**Category : **Mathematics

**Languages : **en

**Pages : **341

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**Book Description**

**Author**: Martin W. Liebeck

**Publisher:** American Mathematical Soc.

**ISBN:** 0821824945

**Category : **Mathematics

**Languages : **en

**Pages : **151

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**Book Description**
Factorizations of finite groups as a product of two proper subgroups arise naturally in several areas of group theory, geometry, and applications. In this book, the authors determine all factorizations of the finite simple groups and their automorphism groups as a product of two maximal subgroups. The proof involved detailed study of the geometry of simple groups, and there is a substantial introductory section presenting this material.

**Author**: Michael Aschbacher

**Publisher:** American Mathematical Soc.

**ISBN:** 0821853368

**Category : **Mathematics

**Languages : **en

**Pages : **347

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**Book Description**
The book 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". However, this book provides much more. Chapter 0 is a modern overview of the logical structure of the entire classification. Chapter 1 is a concise but complete outline of the "odd case" with updated references, while Chapter 2 sets the stage for the remainder of the book with a similar outline of the "even case". The remaining six chapters describe in detail the fundamental results whose union completes the proof of the classification theorem. Several important subsidiary results are also discussed. In addition, there is a comprehensive listing of the large number of papers referenced from the literature. Appendices provide a brief but valuable modern introduction to many key ideas and techniques of the proof. Some improved arguments are developed, along with indications of new approaches to the entire classification--such as the second and third generation projects--although there is no attempt to cover them comprehensively. The work should appeal to a broad range of mathematicians--from those who just want an overview of the main ideas of the classification, to those who want a reader's guide to help navigate some of the major papers, and to those who may wish to improve the existing proofs.

**Author**: Daniel Gorenstein

**Publisher:** American Mathematical Soc.

**ISBN:** 082184069X

**Category : **Finite simple groups

**Languages : **en

**Pages : **344

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**Book Description**
The classification of finite simple groups is a landmark result of modern mathematics. The multipart series of monographs which is being published by the AMS (Volume 40.1–40.7 and future volumes) represents the culmination of a century-long project involving the efforts of scores of mathematicians published in hundreds of journal articles, books, and doctoral theses, totaling an estimated 15,000 pages. This part 7 of the series is the middle of a trilogy (Volume 40.5, Volume 40.7, and forthcoming Volume 40.8) treating the Generic Case, i.e., the identification of the alternating groups of degree at least 13 and most of the finite simple groups of Lie type and Lie rank at least 4. Moreover, Volumes 40.4–40.8 of this series will provide a complete treatment of the simple groups of odd type, i.e., the alternating groups (with two exceptions) and the groups of Lie type defined over a finite field of odd order, as well as some of the sporadic simple groups. In particular, this volume completes the construction, begun in Volume 40.5, of a collection of neighboring centralizers of a particularly nice form. All of this is then applied to complete the identification of the alternating groups of degree at least 13. The book is suitable for graduate students and researchers interested in the theory of finite groups.