The Classification of the Finite Simple Groups, Number 7

362 Pages·2018·3.759 MB·English

by  Daniel Gorenstein Richard Lyons Ronald Solomon

M
ATHEMATICAL
Surveys and Monographs
Volume 40, Number 7
The Classification
of the Finite Simple Groups,
Number 7
Daniel Gorenstein
Richard Lyons
Ronald Solomon
The Classification
of the Finite Simple Groups,
Number 7
Part III, Chapters 7–11:
The Generic Case,
Stages 3b and 4a

M
ATHEMATICAL
Surveys and Monographs
Volume 40, Number 7
The Classification
of the Finite Simple Groups,
Number 7
Part III, Chapters 7–11:
The Generic Case,
Stages 3b and 4a
Daniel Gorenstein
Richard Lyons
Ronald Solomon
Editorial Board
Robert Guralnick Benjamin Sudakov
Michael A. Singer, Chair ConstantinTeleman
Michael I. Weinstein
The authors gratefully acknowledge the support provided by grants from the Na-
tionalSecurityAgency(H98230-07-1-0003andH98230-13-1-0229),theSimonsFoundation
(425816),and the Ohio State University Emeritus Academy.
2010 Mathematics Subject Classification. Primary 20D06, 20D08;
Secondary 20D05, 20E32,20G40.
For additional informationand updates on this book, visit
www.ams.org/bookpages/surv-40.7
The ISBN numbers for this series of books includes
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. (Mathematicalsurveysandmonographs: v.40,number1–)
Includesbibliographicalreferencesandindex.
ISBN0-8218-0334-4[number1]
1. Finitesimplegroups. I.Lyons,Richard,1945–. II.Solomon,Ronald. III.Title. IV.Series:
Mathematicalsurveysandmonographs,no.40,pt. 1–;.
QA177.G67 1994
512(cid:2).2-dc20 94-23001
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Contents
Preface ix
Chapter 7. The Stages of Theorem C∗ 1
7
1. Introduction 1
2. The Ranking Function F 6
3. The Set J∗(G), and the Four Cases 7
p
4. The Sets J1(G) and J2(G), and How to Modify Stages 2 and 3a 8
p p
5. Component Pairs 11
6. The Remaining Stages of Theorem C∗ 12
7
Chapter 8. General Group-Theoretic Lemmas 19
1. p-Groups 19
2. Pumpups and p-Terminality 20
3. Balance and Signalizers 27
4. Components and Coverings 28
5. Representations and Cohomology 29
6. Fusion and Transfer 31
7. Miscellaneous 35
Chapter 9. Theorem C∗: Stage 3b 39
7
1. Introduction 39
2. Theorem 2: The Setup and Special Cases 41
∼ ∼
3. The Cases K =A or K =He, with p=2 45
n
4. The F Case 54
5
5. p-Thin Configurations, p Odd 56
6. The General Lie Type Case, Begun 63
7. The General Lie Type Case, Concluded 75
8. The p=2, SL (q) case 88
2
9. Theorem 3 95
10. Theorem 4: The Prime s 102
11. The s-Uniqueness Case 111
12. Theorem 5: The General Case 115
13. Theorem 5: Residual Cases, p=2 121
14. Theorem 5: Residual Cases, p>2 126
15. Theorem 6 143
Chapter 10. Theorem C∗, Stage 4a: Constructing a Large Alternating
7
Subgroup G 145
0
1. Introduction 145
2. Theorems 1 and 2: Most Sporadic Components 147
vii
viii CONTENTS
3. Theorems 1 and 2: J and Ree Components 149
1
4. The Alternating Case: Theorem 3–Centralizers of Rank 2 152
5. Theorem 4–Centralizers of Rank 1: Preliminaries 157
6. Theorem 4–All Alternating Group Neighbors 160
7. Theorem 4–Non-Alternating Group Neighbors 167
8. Theorem 4–The Final Case: 2HS Neighbors 175
Chapter 11. Properties of K-Groups 181
1. Everyday Tools 181
2. Automorphisms and Coverings 189
3. p-Rank and p-Sylow Structure of Quasisimple Groups 191
4. Computations in Alternating Groups 196
5. Computations in Sporadic Groups 198
6. Computations in Groups of Lie Type 203
7. Small Groups 217
8. Embeddings 222
9. Recognition Theorems 223
10. Balance 223
11. Subcomponents 224
12. Pumpups and the Ranking Function F 226
13. Pumpups 231
14. Acceptable Subterminal Pairs, I 262
15. Acceptable Subterminal Pairs, II 271
16. Neighborhoods 277
17. Generation 288
18. Splitting Primes 292
19. Double Pumpups 298
20. The Double Pumpup Propositions 303
21. Monotonicity of F, and J2(G) 320
p
22. Miscellaneous 337
Bibliography 341
Index 343
Preface
Volumes 5, 7, and 8 of this series form a trilogy treating the Generic Case of
the classification proof, of which this is the middle volume. An overview of the
general strategy for this set of volumes, along with a brief history of the original
treatmentoftheseresults,isprovidedintheprefacetoVolume5,towhichwerefer
the reader.
Using the Signalizer Functor Method, we have arrived by the end of Volume
5 at the existence in our K-proper simple group G, for a suitably chosen prime p,
of elements x of order p whose centralizer has a generic quasi-simple component
K with C (K) of very small p-rank. Moreover, either p = 2 or K ∈ Chev(2).
G
Significantly, also, a family of “neighboring” centralizers have semisimple p-layers.
A precise statement of the initial conditions for this volume is given as Theorem
1.2 in Chapter 7 of this volume. [Note: The numbering of chapters in this volume
continues that of Volume 5. In particular, Chapter 7 is the first chapter of this
volume.]
The principal theorem established in this volume is Theorem C∗: Stage 3b,
7
which is the subject of Chapter 9 and is stated carefully in the introduction to
that chapter. One of the main conclusions (Theorem 2) of this theorem is that
the neighborhood N of (x,K) is vertical. The adjective “vertical” is in contrast
to “diagonal”. This theorem rules out the “shadow” possibility that G=K (cid:5)(cid:6)x(cid:7),
1
whereK isdiagonallyembeddedinthebaseK ×···×K ofthewreathedproduct.
1 p
Of course, this structure would violate the assumed simplicity of G, but it must be
ruled out from local information.
The remaining parts of Theorem C∗: Stage 3b address the structure of K and
7
its “neighbors” when K is a group of Lie type. In the process we define a total
ordering on the set of all groups of Lie type. This ordering is compatible with a
naturalpartialorderingoncomponentsderivingfromL-balanceandthe“pumping-
up” process. Most of our results are proved only for those K which are maximal in
thetotalordering. First, weestablishinTheorem4that,byshiftingprimesifneed
be, we may assume that p splits K, i.e., if K is defined over a field of order q, then
p divides q2−1. This is, of course, trivial if p = 2 and q is odd. The significance
of this result is to make a good choice of the odd prime p when q is a power of 2.
Once we have chosen p to split K, the last major result (Theorem 5) shows
that,withfewexceptions,theneighborhoodNislevel. Essentially, thismeansthat
thecomponent L ofC (y)isdefinedoverthesame fieldasisK, whenever(y,L )
y G y
is a neighbor of (x,K). The “shadow” lurking behind this case is G = K(qp)(cid:6)x(cid:7)
with x acting as a field or graph-field automorphism on E(G)= K(qp). Again, of
course, this structure violates the simplicity of G. Usually, a contradiction follows
easily from the ordering we have placed on components, but sometimes a transfer
argument is used to show that x(cid:8)∈[G,G], violating the assumed simplicity of G.
ix