1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

2.5. ELIMINATING THE·SHADOWS WITH rQ EMPTY 567 Suppose first that i either centralizes Kx or induces an outer automorphism on Kx ...

568 2. CLASSIFYING THE GROUPS WITH [M(T)[ =^1 PROOF. This follows from 2.5.12.3. D LEMMA 2.5.24. <I>(Sc) =/:-1. PROOF. Ass ...

2.5. ELIMINATING THE SHADOWS WITH rg EMPTY 569 transposition on Ki· As Cs 0 (i) = (i, t), Sc is dihedral or semidihedral by a le ...

570 2. CLASSIFYING THE GROUPS WITH JM(T)J = 1 hence e inverts vx and centralizes Sa. Therefore ex inverts V and centralizes S(;, ...

CHAPTER 3 Determining the cases for L E £j ( G, T) By Theorem 2.1.1, we may assume in the remainder of the proof of our Main The ...

572 3. DETERMINING THE CASES FOR LE .C'j (G, T) R :SJ Mo and R E Syb(0^2 (H)R), so C.5.1.2 holds. Thus Hypothesis C.5.1 is indee ...

3.1. COMMON NORMAL SUBGROUPS, AND THE qrc-LEMMA FOR QTKE-GROUPS 573 HYPOTHESIS 3.1.5. T:::; Mo:::; M, HE H*(T,M), and VE R 2 (M ...

574 3. DETERMINING THE CASES FOR LE C'j(G, T) Hypothesis 3.1.5, V E R 2 (M 0 ), so V is elementary abelian, normal in T, and con ...

3.1. COMMON NORMAL SUBGROUPS, AND THE qrc-LEMMA FOR QTKE-GROUPS 575 Let Y denote a Hall 2' -subgroup of B. As D :::) B by the pr ...

S76 3. DETERMINING THE CASES FOR LE .Lj (G, T) HE He, so UH E R2(H) by B.2.14. As Ki CH(UH), CH(UH) :::; kerMnH(H) by B.6.8.6, a ...

3.1. COMMON NORMAL SUBGROUPS, AND THE qrc-LEMMA FOR QTKE-GROUPS 577 2m(B/CB(Vh)) = m(Vh/Cvn(B)) for each h EH with [Vh,B] =f. 1, ...

578 3. DETERMINING THE CASES FOR LE .C.j(G, T) A.l.31.1, s :::; 2; by E.2.3.2, if Y/ is a symmetric group, then Y/ is 83 or 85; ...

3.2. THE FUNDAMENTAL SETUP, AND THE CASE DIVISION FOR .Cj(G, T) 579 HYPOTHESIS 3.2.1 (Fundamental Setup (FSU)). G is a simple QT ...

580 3. DETERMINING THE CASES FOR LE L:.j(G, T) Irr +(L 0 , R 2 (L 0 T)) there. exists Vo E Irr +(Lo, R2(LoT), T) with Vo/CvJLo) ...

3.2. THE FUNDAMENTAL SETUP, AND THE CASE DIVISION FOR .Cj(G, T) 581 may assume that Cv 0 (L) = 0. Then by 3.2.2.5, Vo is a TI-se ...

582 3. DETERMINING THE CASES FOR LE .Cj(G, T) V = VM or Vis a TI-set under M, by 3.2.5. Thus Lo and V are described in 3.2.6, wh ...

3.2. THE FUNDAMENTAL SETUP, AND THE CASE DIVISION FOR C.j(G, T) 583 automophism nontrivial on the Dynkin diagram. Finally we eli ...

584 3. DETERMINING THE CASES FOR LE .C.j(G, T) abelian, and thus (5) implies (6) in this case. Now assume the hypothesis of (6b) ...

3.3. NORMALIZERS OF UNIQUENESS GROUPS CONTAIN Na(T) 585 so D1(Z(R)) is 2-reduced. Therefore R 2 (XT) = D 1 (Z(R)), so H acts on ...

586 3. DETERMINING THE CASES FOR LE .Cj(G, T) In this section, in Theorem 3.3.1 we establish an important property of maximal 2- ...