DOCSLIB.ORG

View This Volume's Front and Back Matter

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
View This Volume's Front and Back Matter MATHEMATICAL Surveys and Monographs Volume 40, Number 8 The Classification of the Finite Simple Groups, Number 8 Daniel Gorenstein Richard Lyons Ronald Solomon 10.1090/surv/040.8 The Classification of the Finite Simple Groups, Number 8 Part III, Chapters 12 –17: The Generic Case, Completed MATHEMATICAL Surveys and Monographs Volume 40, Number 8 The Classification of the Finite Simple Groups, Number 8 Part III, Chapters 12 –17: The Generic Case, Completed Daniel Gorenstein Richard Lyons Ronald Solomon Editorial Board Robert Guralnick Benjamin Sudakov Michael A. Singer, Chair Constantin Teleman MichaelI.Weinstein The authors gratefully acknowledge the support provided by grants from the Na- tional Security Agency (H98230-07-1-0003 and H98230-13-1-0229), the Simons Foundation (425816), and the Ohio State University Emeritus Academy. 2010 Mathematics Subject Classification. Primary 20D05, 20D06, 20D08; Secondary 20E25, 20E32, 20F05, 20G40. For additional information and updates on this book, visit www.ams.org/bookpages/surv-40.8 The ISBN numbers for this series of books includes ISBN 978-1-4704-4189-0 (number 8) ISBN 978-0-8218-4069-6 (number 7) ISBN 978-0-8218-2777-2 (number 6) ISBN 978-0-8218-2776-5 (number 5) ISBN 978-0-8218-1379-9 (number 4) ISBN 978-0-8218-0391-2 (number 3) ISBN 978-0-8218-0390-5 (number 2) ISBN 978-0-8218-0334-9 (number 1) Library of Congress Cataloging-in-Publication Data The first volume was catalogued as follows: Gorenstein, Daniel. The classification of the finite simple groups / Daniel Gorenstein, Richard Lyons, Ronald Solomon. p. cm. (Mathematical surveys and monographs: v. 40, number 1–) Includes bibliographical references and index. ISBN 0-8218-0334-4 [number 1] 1. Finite simple groups. I. Lyons, Richard, 1945– . II. Solomon, Ronald. III. Title. IV. Series: Mathematical surveys and monographs, no. 40, pt. 1–;. QA177 .G67 1994 512.2-dc20 94-23001 CIP Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. c 2018 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10987654321 232221201918 For Sofia and Larry Contents Preface ix Chapter 12. Introduction 1 C C∗ 1. Theorems 7 and 7 1 Chapter 13. Recognition Theory 9 1. Curtis-Tits Systems and Phan Systems 9 2. The Gilman-Griess Theorem for Groups in Chev(2) 14 3. The Wong-Finkelstein-Solomon Method 15 C∗ Chapter 14. Theorem 7: Stage 4b+. A Large Lie-Type Subgroup G0 for p =2 29 1. Introduction 29 ± 2. A 2-Local Characterization of L4 (q), q Odd 33 ∼ ± 3. The Case K = L3 (q)39 ∼ 3 4. The Case K = G2(q)or D4(q)41 5. The Non-Level Case 47 6. The Other Exceptional Cases 58 ∼ a 7. The Case K = PSp2n(q)=Cn(q) 62 8. The Spinn(q)Cases,n ≥ 774 9. The Sp2n Cases 80 10. The Linear and Unitary Cases 87 11. The Orthogonal Case, Preliminaries 112 12. The Orthogonal Case, Completed 122 13. The Cases in Which G0 is Exceptional 141 14. Summary: p = 2 154 C∗ Chapter 15. Theorem 7: Stage 4b+. A Large Lie-Type Subgroup G0 for p>2 155 1. Introduction 155 2. A Choice of p 156 3. The Weyl Group 157 4. The Field Automorphism Case 160 5. Some General Lemmas 167 6. The Case m (B) = 4 178 p ∼ 7. The Case Aut (B) = W (BC )orW (F ) 187 K ∼ n 4 8. The Case AutK (B) = W (Dn), n ≥ 4 193 9. Some Exceptional Cases 206 ∼ ± 10. The Case K/Z(K) = PSLm(q) 207 ∼ 11. The Final Case: K/Z(K) = E6(q) 219 vii viii CONTENTS 12. Identification of G0: Setup 224 ∼ −q 13. G0 = Sp2n+2(q)orAn+1(q) 225 ∼ ± 14. G0 = Dn+1(q) 232 ∼ q 15. G = L (q) 248 0 ∼ k 16. G0 = E8(q) 252 ∼ q 17. G0 = E6 (q)andE7(q) 257 18. The Remaining Cases for G0 264 19. ΓD,1(G) Normalizes G0 277 C∗ Chapter 16. Theorem 7: Stage 5+. G = G0 291 1. Introduction and Generalities 291 2. The Alternating Case 293 3. The Lie Type Case, p = 2: Part 1 294 4. The Lie Type Case, p = 2: Part 2 298 5. The Lie Type Case, p>2 303 Chapter 17. Preliminary Properties of K-Groups 321 1. Weyl Groups and Their Representations 321 2. Toral Subgroups 335 3. Neighborhoods 355 4. CTP-Systems 376 5. Representations 383 6. Computations in Groups of Lie Type 387 7. Outer Automorphisms, Covering Groups, and Envelopes 406 8. p-Structure of Quasisimple K-groups 408 9. Generation 415 10. Pumpups 423 11. Small Groups 455 12. Subcomponents 464 13. Acceptable Subterminal Pairs 470 14. Fusion 474 15. Balance and Signalizers 480 16. Miscellaneous 481 Bibliography 485 Index 487 Preface Volumes 5, 7, and 8 of this series form a trilogy treating the Generic Case of the classification proof. An overview of the general strategy for this set of volumes, along with a brief history of the original treatment of these results, is provided in the preface and Chapter 1 of Volume 5. We shall not repeat that here; rather, we refer the reader to Volume 5. By the end of Volume 7, we arrived at the existence in our K-proper simple group G, for a suitably chosen prime p, of one of the following: ∼ (a) (Alternating case) a subgroup G0 ≤ G such that G0 = An for some n ≥ 13, p =2,andΓD,1(G) normalizes G0 for any root 4-subgroup D of G0. Thus for any involution d ∈ G0 which is the product of two disjoint transpositions, CG(d) normalizes G0;or (b) (Lie-type case) an element x of order p whose centralizer CG(x)hasa p-generic quasisimple component K = dL(q) ∈ Chev(r), where p, r are a pair of distinct primes with q apowerofr, such that either p =2andr is odd, or r =2andp divides q2 − 1. In Chapter 16 of this volume, we shall prove that in case (a), G = G0,sothat G is a K-group, as desired. However, almost all our attention in this volume will be on case (b), where in order to catch up with case (a), we construct a subgroup G ∈ Chev such that Γ (G):=N (Q) | 1 = Q ≤ D normalizes G for a suitable 0 ∼ D,1 G 0 subgroup D = Zp × Zp of G. The results of Volume 7 provide a lot more information in case (b). Thus, we know that CG(K) has a cyclic or quaternion Sylow p-subgroup, and if p> 2, then K itself contains a copy of Zp × Zp × Zp. Significantly, also, a family of centralizers “neighboring” CG(x) also have semisimple p-layers. Considerable additional information is known about these centralizers. All this is spelled out precisely in Theorem 1.2 in Chapter 12 of this volume. [Note: The numbering of chapters in this volume continues that of Volumes 5 and 7. In particular, Chapter 12 is the first chapter of this volume.] Roughly speaking, using the terminology of the initial treatment of the Classification Theorem, we are faced with “standard form problems” for components in Chev whose centralizers have p-rank 1. In this volume we complete the proof of C∗ K ∅ Theorem 7. Let G be a -proper simple group. Assume that γ(G) = and that G does not possess a p-Thin Configuration for any prime ∼ (7)∗ p ∈ γ(G).ThenG = G0 for some G0 ∈ K . ∗ (See p. 1 and p. 4 for the definitions of the set K(7) of “generic” known simple groups and the set γ(G) of primes associated to G.) In particular, in conjunction with the 2-Uniqueness Theorems in Volume 4 and the main theorems of Volume 6, ix xPREFACE we have completed our treatment of simple groups of odd type, in the sense that we have proved the following theorem. Theorem O. Let G be a K-proper simple group. Assume that either m2(G) ≤ Lo ∩ T ∪ G ∅ ∈ C 2 or 2(G) ( 2 2) = . Then either G is an alternating group or G hev(r) ∼ for some odd prime r or G = M11, M12, J1, Mc, Ly,orO N. Put another way, we have the following theorem. Theorem. Let G be a minimal counterexample to the Classification Theorem. Then G is of even type, i.e., the following conditions hold: (a) m2(G) ≥ 3; (b) O2 (CG(t)) = 1 for all t ∈I2(G);and Lo ⊆ C (c) 2(G) 2. C∗ Theorem 7 also includes the following result when G is of even type. Theorem GE. Let G be a K-proper simple group of even type.
Load more

Recommended publications
  • The Thompson-Lyons Transfer Lemma for Fusion Systems
    Submitted exclusively to the London Mathematical Society doi:10.1112/0000/000000 The Thompson-Lyons transfer lemma for fusion systems Justin Lynd Abstract A generalization of the Thompson transfer lemma and its various extensions, most recently due to Lyons, is proven in the context of saturated fusion systems. A strengthening of Alperin’s fusion theorem is also given in this setting, following Alperin’s own “up and down” fusion. The classical Thompson transfer lemma appeared as Lemma 5.38 in [12]; for a 2-perfect group G with S ∈ Syl2(G), it says that if T is a maximal subgroup of S and u is an involution in S − T , then the element u has a G-conjugate in T . Thompson’s lemma has since been generalized in a number of ways. Harada showed [9, Lemma 16] that the same conclusion holds provided one takes u to be of least order in S − T . Unpublished notes of Goldschmidt [6] extended this to show that one may find an G-conjugate of u in T which is extremal under the same conditions. An element t ∈ S is said to be extremal in S with respect to G if CS(t) is a Sylow subgroup of CG(t). In other words, hti is fully FS(G)-centralized, where FS(G) is the fusion system of G. Later, Thompson’s result and its extensions were generalized to all primes via an argument of Lyons [7, Proposition 15.15]. We prove here a common generalization of Lyons’ extension and his similar transfer result [8, Chapter 2, Lemma 3.1] (which relaxes the requirement that S/T be cyclic) in the context of saturated fusion systems.
    [Show full text]
  • Daniel Gorenstein 1923-1992
    Daniel Gorenstein 1923-1992 A Biographical Memoir by Michael Aschbacher ©2016 National Academy of Sciences. Any opinions expressed in this memoir are those of the author and do not necessarily reflect the views of the National Academy of Sciences. DANIEL GORENSTEIN January 3, 1923–August 26, 1992 Elected to the NAS, 1987 Daniel Gorenstein was one of the most influential figures in mathematics during the last few decades of the 20th century. In particular, he was a primary architect of the classification of the finite simple groups. During his career Gorenstein received many of the honors that the mathematical community reserves for its highest achievers. He was awarded the Steele Prize for mathemat- New Jersey University of Rutgers, The State of ical exposition by the American Mathematical Society in 1989; he delivered the plenary address at the International Congress of Mathematicians in Helsinki, Finland, in 1978; Photograph courtesy of of Photograph courtesy and he was the Colloquium Lecturer for the American Mathematical Society in 1984. He was also a member of the National Academy of Sciences and of the American By Michael Aschbacher Academy of Arts and Sciences. Gorenstein was the Jacqueline B. Lewis Professor of Mathematics at Rutgers University and the founding director of its Center for Discrete Mathematics and Theoretical Computer Science. He served as chairman of the universi- ty’s mathematics department from 1975 to 1982, and together with his predecessor, Ken Wolfson, he oversaw a dramatic improvement in the quality of mathematics at Rutgers. Born and raised in Boston, Gorenstein attended the Boston Latin School and went on to receive an A.B.
    [Show full text]
  • The Classification of the Finite Simple
    http://dx.doi.org/10.1090/surv/040.4 Selected Titles in This Series 68 David A. Cox and Sheldon Katz, Mirror symmetry and algebraic geometry, 1999 67 A. Borel and N. Wallach, Continuous cohomology, discrete subgroups, and representations of reductive groups, Second Edition, 1999 66 Yu. Ilyashenko and Weigu Li, Nonlocal bifurcations, 1999 65 Carl Faith, Rings and things and a fine array of twentieth century associative algebra, 1999 64 Rene A. Carmona and Boris Rozovskii, Editors, Stochastic partial differential equations: Six perspectives, 1999 63 Mark Hovey, Model categories, 1999 62 Vladimir I. Bogachev, Gaussian measures, 1998 61 W. Norrie Everitt and Lawrence Markus, Boundary value problems and symplectic algebra for ordinary differential and quasi-differential operators, 1999 60 Iain Raeburn and Dana P. Williams, Morita equivalence and continuous-trace C*-algebras, 1998 59 Paul Howard and Jean E. Rubin, Consequences of the axiom of choice, 1998 58 Pavel I. Etingof, Igor B. Frenkel, and Alexander A. Kirillov, Jr., Lectures on representation theory and Knizhnik-Zamolodchikov equations, 1998 57 Marc Levine, Mixed motives, 1998 56 Leonid I. Korogodski and Yan S. Soibelman, Algebras of functions on quantum groups: Part I, 1998 55 J. Scott Carter and Masahico Saito, Knotted surfaces and their diagrams, 1998 54 Casper Goffman, Togo Nishiura, and Daniel Waterman, Homeomorphisms in analysis, 1997 53 Andreas Kriegl and Peter W. Michor, The convenient setting of global analysis, 1997 52 V. A. Kozlov, V. G. Maz'ya, and J. Rossmann, Elliptic boundary value problems in domains with point singularities, 1997 51 Jan Maly and William P. Ziemer, Fine regularity of solutions of elliptic partial differential equations, 1997 50 Jon Aaronson, An introduction to infinite ergodic theory, 1997 49 R.
    [Show full text]
  • Prize Is Awarded Every Three Years at the Joint Mathematics Meetings
    AMERICAN MATHEMATICAL SOCIETY LEVI L. CONANT PRIZE This prize was established in 2000 in honor of Levi L. Conant to recognize the best expository paper published in either the Notices of the AMS or the Bulletin of the AMS in the preceding fi ve years. Levi L. Conant (1857–1916) was a math- ematician who taught at Dakota School of Mines for three years and at Worcester Polytechnic Institute for twenty-fi ve years. His will included a bequest to the AMS effective upon his wife’s death, which occurred sixty years after his own demise. Citation Persi Diaconis The Levi L. Conant Prize for 2012 is awarded to Persi Diaconis for his article, “The Markov chain Monte Carlo revolution” (Bulletin Amer. Math. Soc. 46 (2009), no. 2, 179–205). This wonderful article is a lively and engaging overview of modern methods in probability and statistics, and their applications. It opens with a fascinating real- life example: a prison psychologist turns up at Stanford University with encoded messages written by prisoners, and Marc Coram uses the Metropolis algorithm to decrypt them. From there, the article gets even more compelling! After a highly accessible description of Markov chains from fi rst principles, Diaconis colorfully illustrates many of the applications and venues of these ideas. Along the way, he points to some very interesting mathematics and some fascinating open questions, especially about the running time in concrete situ- ations of the Metropolis algorithm, which is a specifi c Monte Carlo method for constructing Markov chains. The article also highlights the use of spectral methods to deduce estimates for the length of the chain needed to achieve mixing.
    [Show full text]
  • Representations of Finite Groups and Applications
    P. I. C. M. – 2018 Rio de Janeiro, Vol. 2 (241–266) REPRESENTATIONS OF FINITE GROUPS AND APPLICATIONS P H T Abstract We discuss some basic problems in representation theory of finite groups, and cur- rent approaches and recent progress on some of these problems. We will also outline some applications of these and other results in representation theory of finite groups to various problems in group theory, number theory, and algebraic geometry. 1 Introduction Let G be a finite group and F be a field. A (finite-dimensional) representation of G over F is a group homomorphism Φ: G GL(V ) for some finite-dimensional vector space ! V over F. Such a representation Φ is called irreducible if 0 and V are the only Φ(G)- f g invariant subspaces of V . Representation theory of finite groups started with the letter correspondence between Richard Dedekind and Ferdinand Georg Frobenius in April (12th, 17th, 26th) 1896. In the same year, Frobenius constructed the character table of PSL2(p), p any prime. Later on, the foundations of the complex representation theory (i.e. when F = C), were developed by Frobenius, Dedekind, Burnside, Schur, Noether, and others. Foundations of the modu- lar representation theory (that is, when p = char(F) > 0 and p divides G ) were (almost j j singlehandedly) laid out by Richard Brauer, started in 1935 and continued over the next few decades. A natural question arises: in a more-than-century-old theory such as the representa- tion theory of finite groups, what could still remain to be studied? By the Jordan–Hölder theorem, irreducible representations are the building blocks of any finite-dimensional rep- resentation of any finite group G.
    [Show full text]
  • A Survey: Bob Griess's Work on Simple Groups and Their
    Bulletin of the Institute of Mathematics Academia Sinica (New Series) Vol. 13 (2018), No. 4, pp. 365-382 DOI: 10.21915/BIMAS.2018401 A SURVEY: BOB GRIESS’S WORK ON SIMPLE GROUPS AND THEIR CLASSIFICATION STEPHEN D. SMITH Department of Mathematics (m/c 249), University of Illinois at Chicago, 851 S. Morgan, Chicago IL 60607-7045. E-mail: [email protected] Abstract This is a brief survey of the research accomplishments of Bob Griess, focusing on the work primarily related to simple groups and their classification. It does not attempt to also cover Bob’s many contributions to the theory of vertex operator algebras. (This is only because I am unqualified to survey that VOA material.) For background references on simple groups and their classification, I’ll mainly use the “outline” book [1] Over half of Bob Griess’s 85 papers on MathSciNet are more or less directly concerned with simple groups. Obviously I can only briefly describe the contents of so much work. (And I have left his work on vertex algebras etc to articles in this volume by more expert authors.) Background: Quasisimple components and the list of simple groups We first review some standard material from the early part of [1, Sec 0.3]. (More experienced readers can skip ahead to the subsequent subsection on the list of simple groups.) Received September 12, 2016. AMS Subject Classification: 20D05, 20D06, 20D08, 20E32, 20E42, 20J06. Key words and phrases: Simple groups, classification, Schur multiplier, standard type, Monster, code loops, exceptional Lie groups. 365 366 STEPHEN D. SMITH [December Components and the generalized Fitting subgroup The study of (nonabelian) simple groups leads naturally to consideration of groups L which are: quasisimple:namely,L/Z(L) is nonabelian simple; with L =[L, L].
    [Show full text]
  • Notices of the American Mathematical Society ABCD Springer.Com
    ISSN 0002-9920 Notices of the American Mathematical Society ABCD springer.com Visit Springer at the of the American Mathematical Society 2010 Joint Mathematics December 2009 Volume 56, Number 11 Remembering John Stallings Meeting! page 1410 The Quest for Universal Spaces in Dimension Theory page 1418 A Trio of Institutes page 1426 7 Stop by the Springer booths and browse over 200 print books and over 1,000 ebooks! Our new touch-screen technology lets you browse titles with a single touch. It not only lets you view an entire book online, it also lets you order it as well. It’s as easy as 1-2-3. Volume 56, Number 11, Pages 1401–1520, December 2009 7 Sign up for 6 weeks free trial access to any of our over 100 journals, and enter to win a Kindle! 7 Find out about our new, revolutionary LaTeX product. Curious? Stop by to find out more. 2010 JMM 014494x Adrien-Marie Legendre and Joseph Fourier (see page 1455) Trim: 8.25" x 10.75" 120 pages on 40 lb Velocity • Spine: 1/8" • Print Cover on 9pt Carolina ,!4%8 ,!4%8 ,!4%8 AMERICAN MATHEMATICAL SOCIETY For the Avid Reader 1001 Problems in Mathematics under the Classical Number Theory Microscope Jean-Marie De Koninck, Université Notes on Cognitive Aspects of Laval, Quebec, QC, Canada, and Mathematical Practice Armel Mercier, Université du Québec à Chicoutimi, QC, Canada Alexandre V. Borovik, University of Manchester, United Kingdom 2007; 336 pages; Hardcover; ISBN: 978-0- 2010; approximately 331 pages; Hardcover; ISBN: 8218-4224-9; List US$49; AMS members 978-0-8218-4761-9; List US$59; AMS members US$47; Order US$39; Order code PINT code MBK/71 Bourbaki Making TEXTBOOK A Secret Society of Mathematics Mathematicians Come to Life Maurice Mashaal, Pour la Science, Paris, France A Guide for Teachers and Students 2006; 168 pages; Softcover; ISBN: 978-0- O.
    [Show full text]
  • Collaborative Mathematics and the 'Uninvention' of a 1000-Page Proof
    436547SSS Social Studies of Science 42(2) 185 –213 A group theory of group © The Author(s) 2012 Reprints and permission: sagepub. theory: Collaborative co.uk/journalsPermissions.nav DOI: 10.1177/0306312712436547 mathematics and the sss.sagepub.com ‘uninvention’ of a 1000-page proof Alma Steingart Program in History, Anthropology, and Science, Technology, and Society, MIT, Cambridge, MA, USA Abstract Over a period of more than 30 years, more than 100 mathematicians worked on a project to classify mathematical objects known as finite simple groups. The Classification, when officially declared completed in 1981, ranged between 300 and 500 articles and ran somewhere between 5,000 and 10,000 journal pages. Mathematicians have hailed the project as one of the greatest mathematical achievements of the 20th century, and it surpasses, both in scale and scope, any other mathematical proof of the 20th century. The history of the Classification points to the importance of face-to-face interaction and close teaching relationships in the production and transformation of theoretical knowledge. The techniques and methods that governed much of the work in finite simple group theory circulated via personal, often informal, communication, rather than in published proofs. Consequently, the printed proofs that would constitute the Classification Theorem functioned as a sort of shorthand for and formalization of proofs that had already been established during personal interactions among mathematicians. The proof of the Classification was at once both a material artifact and a crystallization of one community’s shared practices, values, histories, and expertise. However, beginning in the 1980s, the original proof of the Classification faced the threat of ‘uninvention’.
    [Show full text]
  • The Classification of Finite Simple Groups: a Progress Report
    The Classification of Finite Simple Groups: A Progress Report Ronald Solomon Communicated by Alexander Diaz-Lopez and Harriet Pollatsek Danny Gorenstein dubbed the classification of finite simple groups “The Thirty Years’ War.” Ronald Solomon is professor emeritus at the Ohio State University. His email address is [email protected]. For permission to reprint this article, please contact: [email protected]. DOI: http://dx.doi.org/10.1090/noti1689 646 Notices of the AMS Volume 65, Number 6 History of the Project is one significant change worthy of note. (A detailed In 1981 the monumental project to classify all of the fi- overview of the original Classification Project may be nite simple groups appeared to be nearing its conclusion. found in [ALSS].) Danny Gorenstein had dubbed the project the “Thirty We anticipate that there will be twelve volumes in the Years’ War,” dating its inception from an address by complete series [GLS], which we hope to complete by Richard Brauer at the International Congress of Mathe- 2023. I will report on the current state of our project in maticians in 1954. He and Richard Lyons agreed that it this article. would be desirable to write a series of volumes that would contain the complete proof of this Classification Theorem, Introduction to Classification modulo a short and clearly specified list of background re- A common theme in mathematics is to study a partic- sults. As the existing proof was scattered over hundreds ular mathematical structure and attempt to classify all of journal articles, some of which cited other articles instances of it, e.g., regular polyhedra, distance transitive that were never published, there was a consensus that graphs, and 3-manifolds, among others.
    [Show full text]
  • Walter Feit (1930–2004)
    Walter Feit (1930–2004) Leonard Scott, Ronald Solomon, John Thompson, John Walter, and Efim Zelmanov alter Feit died on July 29, 2004, at “Walter Feit was born in Vienna, Austria, in 1930. age seventy-three after a long ill- In August of 1939 his parents saved his life by ness. He had been on the faculty at placing him on the last train (KinderTransport) Yale for forty years, and the follow- allowed to carry Jewish children out of Austria. He Wing is taken from his obituary at the arrived in England just as the British government Yale website. It summarizes Walter’s achievements was evacuating all children from London. After and stature, and provides touching detail on Wal- being relocated a few times, he settled in a refugee ter’s childhood during the Second World War. hostel in Oxford. In 1943 he won a scholarship to “Professor Feit’s 1963 paper with John G.Thomp- an Oxford technical high school. His teachers were son, ‘Solvability of groups of odd order’, filled an very encouraging, and he recorded that it was at this entire issue of the Pacific Journal of Mathematics time that he became ‘passionately interested in and is widely regarded as the most influential paper mathematics’. ever written on finite group theory. It energized the “In 1946 Walter moved to the United States to field, providing both inspiration and technical tools stay with an aunt and uncle. In a letter recovered for the research that finally culminated in the from the time of his arrival in New York, he complete classification of simple finite groups.
    [Show full text]
  • Jordan Decomposition for the Alperin–Mckay Conjecture
    Bergische Universitat¨ Wuppertal Jordan Decomposition for the Alperin{McKay Conjecture Dissertation zur Erlangung des Doktorgrades der Naturwissenschaften im Fachbereich 4 der Bergischen Universit¨atWuppertal vorgelegt von Lucas Ruhstorfer The PhD thesis can be quoted as follows: urn:nbn:de:hbz:468-20200602-122838-0 [http://nbn-resolving.de/urn/resolver.pl?urn=urn%3Anbn%3Ade%3Ahbz%3A468-20200602-122838-0] DOI: 10.25926/pxey-hd44 [https://doi.org/10.25926/pxey-hd44] Contents 1 Representation theory 13 1.1 Modular representation theory . 13 1.2 Module categories . 14 1.3 The Brauer functor . 15 1.4 Brauer pairs and the Brauer category . 17 1.5 Morita equivalences and splendid Rickard equivalences . 19 1.6 First properties of splendid complexes . 21 1.7 Brauer categories and splendid Rickard equivalences . 23 1.8 Properties of splendid Rickard equivalences . 25 1.9 Lifting Rickard equivalences . 26 1.10 Descent of Rickard equivalences . 28 1.11 Morita equivalences and Clifford theory of characters . 30 1.12 Rickard equivalences for the normalizer . 31 1.13 The Brauer functor and Clifford theory . 33 1.14 The Harris–Kn¨orrcorrespondence . 36 1.15 Splendid Rickard equivalences and Clifford theory . 38 2 Deligne{Lusztig theory and disconnected reductive groups 41 2.1 Disconnected reductive algebraic groups . 41 2.2 `-adic cohomology of Deligne{Lusztig varieties . 42 2.3 Properties of Deligne{Lusztig varieties . 43 2.4 Godement resolutions . 47 2.5 Isogenies . 47 2.6 Duality for connected reductive groups . 48 2.7 Levi subgroups, isogenies and duality . 50 2.8 Rational Lusztig series for connected reductive groups .
    [Show full text]
  • A CHARACTERIZATION of the 2-FUSION SYSTEM of L4(Q)
    A CHARACTERIZATION OF THE 2-FUSION SYSTEM OF L4(q) DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Justin Lynd, B.S. Graduate Program in Mathematics The Ohio State University 2012 Dissertation Committee: Dr. Ronald Solomon, Advisor Dr. Matthew Kahle Dr. Jean-Francois Lafont Dr. Richard Lyons c Copyright by Justin Lynd 2012 ABSTRACT We study saturated fusion systems F on a finite 2-group S with involution cen- tralizer having a unique component on a dihedral group and containing the Baumann 2 subgroup of S. Assuming F = O (F), O2(F) = 1, and the centralizer of the com- ponent is a cyclic 2-group, it is shown that F is uniquely determined as the 2-fusion system of L4(q) for some q ≡ 3 (mod 4). This should be viewed as a contribution to a program recently outlined by Aschbacher for the classification of simple fusion systems at the prime 2. The analogous problem in the classification of finite simple groups of component type (the L2(q), A7 standard component problem) was one of the last to be completed, and was ultimately only resolved in an inductive context with heavy machinery. Thanks primarily to the hypothesis concerning the Baumann subgroup and the absence of cores, our arguments by contrast require only 2-fusion analysis and transfer. We prove a generalization of the Thompson transfer lemma in the context of fusion systems, which is applied often. ii To David Reid Dillon iii ACKNOWLEDGMENTS To: Ronald Solomon, for knowing precisely when to push and when to let be, providing continual support, being a careful mentor and constant advocate, sharing his over- whelming breadth of knowledge and insight, and dedicating large amounts time and effort throughout this project (and others prior); I always knew I made the right choice.
    [Show full text]