The classification of the finite simple groups

著者

• Gorenstein, Daniel
• Lyons, Richard
• Solomon, Ronald

書誌事項

The classification of the finite simple groups

Daniel Gorenstein, Richard Lyons, Ronald Solomon

（Mathematical surveys and monographs, v. 40）

American Mathematical Society, c1994-

• [no. 1]
• no. 2
• no. 3
• no. 4
• no. 5
• no. 6
• no. 7
• no. 8

注記

no. 2: pt. 1, chapter G, general group theory -- no. 3: pt. 1, chapter A, almost simple Κ-groups -- no. 4: pt. 2, chapters 1-4, uniqueness theorems -- no. 5: pt. 3, chapters 1-6, the generic case, stages 1-3a -- no. 6: pt. 4, the special odd case -- no. 7: pt. 3, chapters 7-11, the generic case, stages 3b and 4a -- no. 8: pt. 3, chapters 12-17, the genetic case, completed

Includes bibliographical references and indexes

内容説明・目次

巻冊次

[no. 1] ISBN 9780821803349

内容説明

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.

目次

1: Part I. Chapter 1, Overview. 2: Introduction to the series. 3: The finite simple groups. 4: The structure of finite groups. 5: Classifying simple groups. 6: The background results. 7: Sketch of the simplified proof. 8: Additional comments. 9: Part I. Chapter 2, Outline of Proof. 10: Introduction. 11: The grids. 12: The uniqueness grid. 13: The classification grid: Generic and special simple groups. 14: The classification grid: The stages of the proof. 15: Principal techniques of the proof. 16: Notational conventions. 17: Background references. 18: Expository references. 19: Glossary. 20: Index. Part I. Chapter 1, Overview. Introduction to the series. The finite simple groups. The structure of finite groups. Classifying simple groups. The background results. Sketch of the simplified proof. Additional comments. Part I. Chapter 2, Outline of Proof. Introduction. The grids. The uniqueness grid. The classification grid: Generic and special simple groups. The classification grid: The stages of the proof. Principal techniques of the proof. Notational conventions. Background references. Expository references. Glossary. Index. Part I. Chapter 1, Overview. Introduction to the series. The finite simple groups. The structure of finite groups. Classifying simple groups. The background results. Sketch of the simplified proof. Additional comments. Part I. Chapter 2, Outline of Proof. Introduction. The grids. The uniqueness grid. The classification grid: Generic and special simple groups. The classification grid: The stages of the proof. Principal techniques of the proof. Notational conventions. Background references. Expository references. Glossary. Index

巻冊次

no. 2 ISBN 9780821803905

内容説明

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.

目次

General group theory Background references Expository references Glossary Index.

巻冊次

no. 3 ISBN 9780821803912

内容説明

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.

目次

Some theory of linear algebraic groups The finite groups of Lie type Local subgroups of groups of Lie type. I Local subgroups of groups of Lie type. II The alternating groups and the twenty-six sporadic groups Coverings and embeddings of quasisimple $\mathcal {K}$-groups General properties of $\mathcal {K}$-groups Background references Expository references Errata for numbers 1 and 2 Glossary Index.

巻冊次

no. 4 ISBN 9780821813799

内容説明

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.

目次

General lemmas Strongly embedded subgroups and related conditions on involutions $p$-component uniqueness theorems Properties of $K$-groups Background references Expository references Errata for number 3, Chapter I$_A$: Almost simple $\mathcal K$-groups Glossary Index of terminology.

巻冊次

no. 6 ISBN 9780821827772

内容説明

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.

目次

General introduction to the special odd case General lemmas Theorem $C^*_2$: Stage 1 Theorem $C^*_2$: Stage 2 Theorem $C_2$: Stage 3 Theorem $C_2$: Stage 4 Theorem $C_2$: Stage 5 Theorem $C_3$: Stage 1 Theorem $C_3$: Stages 2 and 3 IV$_K$: Preliminary properties of $K$-groups Background references Expository references Glossary Index.

巻冊次

no. 7 ISBN 9780821840696

内容説明

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.

目次

The stages of theorem $\mathscr{C}^*_7$ General group-theoretic lemmas Theorem $\mathscr{C}^*_7$: Stage 3b Theorem $\mathscr{C}^*_7$, stage 4a: Constructing a large alternating subgroup $G_0$ Properties of $\mathscr{K}$-groups Bibliography Index

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詳細情報

• NII書誌ID(NCID): BA23942263
• ISBN:
  • 0821803344
  • 0821803905
  • 0821803913
  • 082181379X
  • 0821827766
  • 0821827774
  • 9780821840696
  • 9781470441890
• LCCN: 94023001
• 出版国コード: us
• タイトル言語コード: eng
• 本文言語コード: eng
• 出版地: Providence, R.I.
• ページ数/冊数: v.
• 大きさ: 26 cm
• 分類:
  • NDC9 : 411.63
  • NDC8 : 411.63
  • LCC : QA177
  • DC20 : 512/.2
• 件名:
  • LCSH : Finite simple groups
• 親書誌ID:
  • BA00134419