طبقه‌بندی گروه‌های سادهٔ متناهی، جلد دوم، بخش اول، فصل جی: نظریهٔ عمومی گروه‌ها

The classification of the finite simple groups is one of the major feats of contemporary mathematical research, but its proof has never been completely extricated from the journal literature in which it first appeared. This book serves as an introduction to a series devoted to organizing and simplifying the proof. The purpose of the series is to present as direct and coherent a proof as is possible with existing techniques. This first volume, which sets up the structure for the entire series, begins with largely informal discussions of the relationship between the Classification Theorem and the general structure of finite groups, as well as the general strategy to be followed in the series and a comparison with the original proof. Also listed are background results from the literature that will be used in subsequent volumes. Next, the authors formally present the structure of the proof and the plan for the series of volumes in the form of two grids, giving the main case division of the proof as well as the principal milestones in the analysis of each case. Thumbnail sketches are given of the ten or so principal methods underlying the proof. This book is intended for first- or second-year graduate students/researchers in group theory.

this Volume Contains The Proofs Of Theorems C2 And C3, Which Constitute The Classification Of Finite Simple Groups G Of Special Odd Type. The Special Odd Condition Is Introduced As Representing The Measure Of Smallness For Simple Groups Which Are Not Of Even Type. Incorporated Into The Proof Of Theorem C2 Are The Classifications By Gorenstein-walther And Alperin-brauer- Gorenstein, Among Others, Noted As Major Accomplishments Of The 1960s. This Treatment Employs The Bender Method Rather Than Signalizer Functor Method. The Authors (whose Academic Affiliations Are Not Given) Include Background And Expository References, And A Glossary-cum-index Of Symbols. Annotation ©2004 Book News, Inc., Portland, Or

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the Second Volume Of A Series Devoted To Reorganizing And Simplifying Proof Of The Classification Of The Finite Simple Groups. In A Single Chapter, It Lays The Groundwork For The Forthcoming Analysis Of Finite Simple Groups, Beginning With The Theory Of Components, Layers, And The Generalized Fitting Subgroup, Which Has Been Developed Largely Since Gorenstein's Basic 1968 Text And Is Now Central To Understanding The Structure Of Finite Groups. Suitable As An Auxiliary Text For A Graduate Course In Group Theory. Member Prices Are $35 For Individual And $47 For Institutions. Annotation C. Book News, Inc., Portland, Or (booknews.com)

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. After three introductory volumes on the classification of the finite simple groups, (Mathematical Surveys and Monographs, Volumes 40.1, 40.2, and 40.3), the authors now start the proof of the classification theorem: They begin the analysis of a minimal counterexample $G$ to the theorem. Two fundamental and powerful theorems in finite group theory are examined: the Bender-Suzuki theorem on strongly embedded subgroups (for which the non-character-theoretic part of the proof is provided) and Aschbacher's Component theorem. Included are new generalizations of Aschbacher's theorem which treat components of centralizers of involutions and $p$-components of centralizers of elements of order $p$ for arbitrary primes $p$. This book, with background from sections of the previous volumes, presents in an approachable manner critical aspects of the classification of finite simple groups. Features: Treatment of two fundamental and powerful theorems in finite group theory. Proofs that are accessible and largely self-contained. New results generalizing Aschbacher's Component theorem and related component uniqueness theorems. This book offers a single source of basic facts about the structure of the finite simple groups with emphasis on a detailed description of their local subgroup structures, coverings and automorphisms. The method is by examination of the specific groups, rather than by the development of an abstract theory of simple groups. While the purpose of the book is to provide the background for the proof of the classification of the finite simple groups—dictating the choice of topics—the subject matter is covered in such depth and detail that the book should be of interest to anyone seeking information about the structure of the finite simple groups. This volume offers a wealth of basic facts and computations. Much of the material is not readily available from any other source. In particular, the book contains the statements and proofs of the fundamental Borel-Tits Theorem and Curtis-Tits Theorem. It also contains complete information about the centralizers of semisimple involutions in groups of Lie type, as well as many other local subgroups. 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 are assembling the proof with explanations and references. It is a monumental task. The book, along with background from sections of the previous volumes, presents critical aspects of the classification. Continuing the proof of the classification theorem which began in the previous five volumes (Surveys of Mathematical Monographs, Volumes 40.1.E, 40.2, 40.3, 40.4, and 40.5), in this volume, the authors provide the classification of finite simple groups of special odd type (Theorems $\mathcal{C}_2$ and $\mathcal{C}_3$, as stated in the first volume of the series). The book is suitable for graduate students and researchers interested in group theory. 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 are assembling the proof with explanations and references. It is a monumental task. The book, along with background from sections of the previous volumes, presents critical aspects of the classification. In four prior volumes (Surveys of Mathematical Monographs, Volumes 40.1, 40.2, 40.3, and 40.4), the authors began the proof of the classification theorem by establishing certain uniqueness and preuniqueness results. In this volume, they now begin the proof of a major theorem from the classification grid, namely Theorem $\mathcal{C}_7$. The book is suitable for graduate students and researchers interested in group theory. No. 1. [without Special Title] -- No. 2. Part I, Chapter G, General Group Theory. -- No. 3. Part I, Chapter A, Almost Simple K-groups. -- No. 4. Part Ii, Chapters 1-4: Uniqueness Theorems. -- No. 5. Part Iii, Chapters 1-6: The Generic Case, Stages 1-3a. -- No. 6. Part Iv, The Special Odd Case -- No. 7. Part Iii, Chapters 7-11: The Generic Case, Stages 3b And 4a. -- No. 8. Part Iii, Chapters 12-17: The Generic Case, Completed. Daniel Gorenstein, Richard Lyons, Ronald Solomon. Includes Bibliographical References And Indexes. Also Available In Electronic Format.