The Classification of the Finite Simple Groups, Number 9

The Classification of the Finite Simple Groups, Number 9 PDF

Author: Inna Capdeboscq

Publisher: American Mathematical Society

Published: 2021-02-22

Total Pages: 520

ISBN-13: 1470464373

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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.

The Classification of the Finite Simple Groups, Number 8: Part III, Chapters 12–17: The Generic Case, Completed

The Classification of the Finite Simple Groups, Number 8: Part III, Chapters 12–17: The Generic Case, Completed PDF

Author: Daniel Gorenstein

Publisher: American Mathematical Soc.

Published: 2018-12-12

Total Pages: 488

ISBN-13: 1470441896

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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.

The Classification of the Finite Simple Groups, Number 10

The Classification of the Finite Simple Groups, Number 10 PDF

Author: Inna Capdeboscq

Publisher: American Mathematical Society

Published: 2023-10-23

Total Pages: 587

ISBN-13: 1470475537

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This book is the tenth in a series of volumes whose aim is to provide a complete proof of the classification theorem for the finite simple groups based on a fairly short and clearly enumerated set of background results. Specifically, this book completes our identification of the simple groups of bicharacteristic type begun in the ninth volume of the series (see SURV/40.9). This is a fascinating set of simple groups which have properties in common with matrix groups (or, more generally, groups of Lie type) defined both over fields of characteristic 2 and over fields of characteristic 3. This set includes 11 of the celebrated 26 sporadic simple groups along with several of their large simple subgroups. Together with SURV/40.9, this volume provides the first unified treatment of this class of simple groups.

The Classification of the Finite Simple Groups, Number 7: Part III, Chapters 7–11: The Generic Case, Stages 3b and 4a

The Classification of the Finite Simple Groups, Number 7: Part III, Chapters 7–11: The Generic Case, Stages 3b and 4a PDF

Author: Daniel Gorenstein

Publisher: American Mathematical Soc.

Published: 2018-02-15

Total Pages: 344

ISBN-13: 082184069X

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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.

The Classification of the Finite Simple Groups

The Classification of the Finite Simple Groups PDF

Author: Daniel Gorenstein

Publisher: American Mathematical Soc.

Published: 1994-11-18

Total Pages: 186

ISBN-13: 0821809601

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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. Much of the book is written in an expository style accessible to nonspecialists.

The Classification of the Finite Simple Groups

The Classification of the Finite Simple Groups PDF

Author: Daniel Gorenstein

Publisher:

Published: 2018

Total Pages: 0

ISBN-13: 9781470441890

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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.

The Classification of the Finite Simple Groups, Number 5

The Classification of the Finite Simple Groups, Number 5 PDF

Author: Daniel Gorenstein

Publisher: American Mathematical Soc.

Published: 1994

Total Pages: 488

ISBN-13: 9780821827765

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The fifth volume of the study proves two, and part of the third, of the planned five stages for the generic cast of the classification of finite simple groups. The main result is that either G has a p-uniqueness subgroup for some prime p, or that G has a neighborhood of semisimple subgroups that demonstrate certain properties in common with those in target simple groups G*. All this is preparation for the final stages, which are expected to deduce that G is about the same as G* for some known simple G*. Stay tuned. Perhaps an index will be deemed meet when the final answers are revealed. Annotation copyrighted by Book News, Inc., Portland, OR

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