The Classification Of The Finite Simple Groups Number 7 Part Iii Chapters 7 11 The Generic Case Stages 3b And 4a

Author : Inna Capdeboscq,Daniel Gorenstein,Richard Lyons,Ronald Solomon
Publisher : American Mathematical Society
Page : 520 pages
File Size : 53,9 Mb
Release : 2021-02-22
Category : Mathematics
ISBN : 9781470464370

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The Classification of the Finite Simple Groups, Number 9 by Inna Capdeboscq,Daniel Gorenstein,Richard Lyons,Ronald Solomon Pdf

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,Richard Lyons,Ronald Solomon
Publisher : American Mathematical Soc.
Page : 488 pages
File Size : 54,8 Mb
Release : 2018-12-12
Category : Electronic
ISBN : 9781470441890

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The Classification of the Finite Simple Groups, Number 8: Part III, Chapters 12–17: The Generic Case, Completed by Daniel Gorenstein,Richard Lyons,Ronald Solomon Pdf

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,Daniel Gorenstein,Richard Lyons,Ronald Solomon
Publisher : American Mathematical Society
Page : 587 pages
File Size : 48,5 Mb
Release : 2023-10-23
Category : Mathematics
ISBN : 9781470475536

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The Classification of the Finite Simple Groups, Number 10 by Inna Capdeboscq,Daniel Gorenstein,Richard Lyons,Ronald Solomon Pdf

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.

Author : Daniel Gorenstein,Richard Lyon,Ronald Solomon
Publisher : American Mathematical Soc.
Page : 344 pages
File Size : 42,9 Mb
Release : 2018-02-15
Category : Finite simple groups
ISBN : 9780821840696

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The Classification of the Finite Simple Groups, Number 7: Part III, Chapters 7–11: The Generic Case, Stages 3b and 4a by Daniel Gorenstein,Richard Lyon,Ronald Solomon Pdf

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.

Author : Daniel Gorenstein,Richard Lyons,Ronald Solomon
Publisher : Unknown
Page : 344 pages
File Size : 41,5 Mb
Release : 2018
Category : Electronic
ISBN : 082184069X

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The Classification of the Finite Simple Groups by Daniel Gorenstein,Richard Lyons,Ronald Solomon Pdf

Author : Daniel Gorenstein,Richard Lyons,Ronald Solomon
Publisher : Unknown
Page : 490 pages
File Size : 55,6 Mb
Release : 2002
Category : Finite simple groups
ISBN : UCSC:32106011398077

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The Classification of the Finite Simple Groups: Part III, chapters 1-6: The generic case, stages 1-3a by Daniel Gorenstein,Richard Lyons,Ronald Solomon Pdf

Author : Daniel Gorenstein,Richard Lyons,Ronald M. Solomon
Publisher : American Mathematical Soc.
Page : 186 pages
File Size : 46,9 Mb
Release : 1994-11-18
Category : Mathematics
ISBN : 9780821809600

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The Classification of the Finite Simple Groups by Daniel Gorenstein,Richard Lyons,Ronald M. Solomon Pdf

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.

Author : Daniel Gorenstein,Richard Lyons,Ronald Solomon
Publisher : American Mathematical Soc.
Page : 488 pages
File Size : 51,8 Mb
Release : 1994
Category : Mathematics
ISBN : 0821827766

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The Classification of the Finite Simple Groups, Number 5 by Daniel Gorenstein,Richard Lyons,Ronald Solomon Pdf

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

Author : Daniel Gorenstein,Richard Lyons,Ronald Solomon
Publisher : Unknown
Page : 0 pages
File Size : 41,8 Mb
Release : 2018
Category : Finite simple groups
ISBN : 1470441896

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The Classification of the Finite Simple Groups by Daniel Gorenstein,Richard Lyons,Ronald Solomon Pdf

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.

Author : Daniel Gorenstein,Richard Lyons,Ronald Solomon
Publisher : Unknown
Page : 467 pages
File Size : 43,9 Mb
Release : 2002
Category : Finite simple groups
ISBN : 0821827766

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The Classification of the Finite Simple Groups by Daniel Gorenstein,Richard Lyons,Ronald Solomon Pdf

Author : Inna Capdeboscq,Daniel Gorenstein,Richard Lyons,Ronald Solomon
Publisher : Unknown
Page : 537 pages
File Size : 51,7 Mb
Release : 2024-06-02
Category : Electronic books
ISBN : 1470465612

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The Classification of the Finite Simple Groups, Number 9 by Inna Capdeboscq,Daniel Gorenstein,Richard Lyons,Ronald Solomon Pdf

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.

Author : Daniel Gorenstein
Publisher : Springer Science & Business Media
Page : 493 pages
File Size : 52,8 Mb
Release : 2013-11-22
Category : Mathematics
ISBN : 9781461336853

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The Classification of Finite Simple Groups by Daniel Gorenstein Pdf

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.

Author : Michael Aschbacher,Richard Lyons ,Stephen D. Smith ,Ronald Solomon
Publisher : American Mathematical Soc.
Page : 362 pages
File Size : 42,7 Mb
Release : 2011
Category : Finite simple groups
ISBN : 9780821853368

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The Classification of Finite Simple Groups by Michael Aschbacher,Richard Lyons ,Stephen D. Smith ,Ronald Solomon Pdf

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'.

Author : Daniel Gorenstein,Richard Lyons,Ronald Solomon
Publisher : American Mathematical Soc.
Page : 446 pages
File Size : 55,8 Mb
Release : 1994
Category : Finite simple groups
ISBN : 0821803913

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The Classification of the Finite Simple Groups, Number 3 by Daniel Gorenstein,Richard Lyons,Ronald Solomon Pdf

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

Author : Stephen D. Smith
Publisher : American Mathematical Soc.
Page : 231 pages
File Size : 42,5 Mb
Release : 2018-04-30
Category : Finite simple groups
ISBN : 9781470442910

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Applying the Classification of Finite Simple Groups: A User’s Guide by Stephen D. Smith Pdf

Classification of Finite Simple Groups (CFSG) is a major project involving work by hundreds of researchers. The work was largely completed by about 1983, although final publication of the “quasithin” part was delayed until 2004. Since the 1980s, CFSG has had a huge influence on work in finite group theory and in many adjacent fields of mathematics. This book attempts to survey and sample a number of such topics from the very large and increasingly active research area of applications of CFSG. The book is based on the author's lectures at the September 2015 Venice Summer School on Finite Groups. With about 50 exercises from original lectures, it can serve as a second-year graduate course for students who have had first-year graduate algebra. It may be of particular interest to students looking for a dissertation topic around group theory. It can also be useful as an introduction and basic reference; in addition, it indicates fuller citations to the appropriate literature for readers who wish to go on to more detailed sources.