1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

0.3. AN OUTLINE OF THE PROOF OF THE MAIN THEOREM 487 There have since been new treatments of portions of the N-group problem due ...

488 INTRODUCTION TO VOLUME II which will emerge below) to work with a subgroup H which is minimal subject to T :::; H, H i. M, a ...

0.3. AN OUTLINE OF THE PROOF OF THE MAIN THEOREM 489 In the next subsection 0.3.2, we describe how to obtain a uniqueness subgro ...

490 INTRODUCTION TO VOLUME TI with P the middle node maximal parabolic over T n Go, and H := PT. Then H ~ (L,T) for an LE C(G,T) ...

0.3. AN OUTLINE OF THE PROOF OF THE MAIN THEOREM 491 a "faithful action", write Xf for those XE X such that (fh(Z(0 2 (X))),X) # ...

492 INTRODUCTION TO VOLUME II (l)02((M 0 ,H)) =I-1, so Mo is not a uniqueness subgroup of G. (2) Vi. 02(H) and q(Mo/CM 0 (V), V) ...

0.3. AN OUTLINE OF THE PROOF OF THE MAIN THEOREM 493 In order to discuss these cases in more detail, we need more concepts and n ...

494 INTRODUCTION TO VOLUME II FSU where Vis not an FF-module. The only quasithin example which arises from those cases is J4, bu ...

0.4. AN OUTLINE OF THE PROOF OF THE EVEN TYPE THEOREM 495 Gz. At this point our recognition theorems show that G is G 2 (3), LH3 ...

496 INTRODUCTION TO VOLUME II This fundamental lemma can be used to show first that L :::l Ca(z)-which is very close to showing ...

Part 1 Structure of QTKE-Groups and the Main Case Division ...

See the Introductions to Volumes I and II for terminology used in this overview. In this first Part, we obtain a solution to the ...

CHAPTER 1 Structure and intersection properties of 2-locals In this chapter we show how the structure theory for SQTK-groups fro ...

500 1. STRUCTURE AND INTERSECTION PROPERTIES OF 2-LOCALS We begin by defining some notation. DEFINITION 1.1.2. Set H= Ha:= {H::: ...

1.1. THE COLLECTION 'He 501 PROOF. Assume the hypotheses of (1) and set N := Na(U). Then by hypoth- esis NE He. Now if U :::'.) ...

502 1. STRUCTURE AND INTERSECTION PROPERTIES OF 2-LOCALS (e) L ~ £ 3 (3) or L 2 (p), p a Fermat or Mersenne prime, and z induces ...

1.2. THE SET .C.* (G, T) OF NONSOLVABLE UNIQUENESS SUBGROUPS 503 Next assume L/Z(L) is of Lie type and odd characteristic; then ...

504 1. STRUCTURE AND INTERSECTION PROPERTIES OF 2-LOCALS (3) If L E C(H), then either L :::] H; or [LH[ = 2 and L/02(L) ~ L2(2n) ...

1.2. THE SET .C*(G, T) OF NONSOLVABLE UNIQUENESS SUBGROUPS 505 Next we extend the notation of .C(X, Y) in Definition A.3.10 to o ...

506 1. STRUCTURE AND INTERSECTION PROPERTIES OF 2-LOCALS We will focus primarily on the case where the role of Sis played by T E ...