1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

3.3. NORMALIZERS OF UNIQUENESS GROUPS CONTAIN Na(T) S87 We recall that M+T is a uniqueness subgroup in the language of chapter 1 ...

588 3. DETERMINING THE CASES FOR LE .c:;ca, T) (3) M+ ~ A 5 or A 7 , and [V, M+] is the natural module for M+. (4) M+ ~ SL 3 (2n ...

3.3. NORMALIZERS OF UNIQUENESS GROUPS CONTAIN NG(T) 589 (b) Y/02(Y) ~ L2(4), K/02(K) ~ Ji, D = (Kn D)Nv(Y), and ID : Nn(Y)I = 7. ...

590 3. DETERMINING THE CASES FOR LE .Cj(G, T) an odd prime p. Here T'K is dihedral of order greater than 4 and self-centralizing ...

3.3. NORMALIZERS OF UNIQUENESS GROUPS CONTAIN Na(T) 59i LEMMA 3.3.16. L is not SL3(2n), Sp4(2n), or G 2 (2n) with n > 1. PROO ...

592 3. DETERMINING THE CASES FOR LE Cj(G, T) Definition C.1.31. As Y 2 /0 2 (Y 2 ) ~ £ 3 (2), case (5) of Theorem C.1.32 holds, ...

3.3. NORMALIZERS OF UNIQUENESS GROUPS CONTAIN Na(T) 593 not contain the unipotent radical of the maximal parabolic determined by ...

594 3. DETERMINING THE CASES FOR LE .C.j(G, T) As D does not act on X, there is g E D - Na(X). Set G1 := XT, G2 .- XBT, and G 0 ...

3.3. NORMALIZERS OF UNIQUENESS GROUPS CONTAIN Na(T) 595. ( c) K 0 < K* ~ L3(p) for some odd prime p. In case (b), D* = 1, con ...

596 3. DETERMINING THE OASES FOR LE .Cj(G, T) Next by 3.3.9, and appealing to B.5.1.6 in case (al): (b) In case (al), CL(Z) ~ 83 ...

3.3. NORMALIZERS OF UNIQUENESS GROUPS CONTAIN Na(T) 597 We have Z :S: [V, L] = V by (e), and T i_ L0 2 (LT) by (f), so V = WEB W ...

598 3. DETERMINING THE CASES FOR LE .Cj(G, T) Cz(L). By (**), 031 (Ca(zd)) = Ld and Ld i= Le since ILi > !Lei· Therefore by ( ...

3.3. NORMALIZERS OF UNIQUENESS GROUPS CONTAIN No(T) 599 Finally if m = 1, let Pi, i = 1, 2, denote the maximal parabolics of LT ...

600 3. DETERMINING THE CASES FOR LE .Cj(G, T) hence A(T) = {A, V} is of order 2. Therefore as Dis of odd order, D acts on V, con ...

3.3. NORMALIZERS OF UNIQUENESS GROUPS CONTAIN Na(T) 601 LEMMA 3.3.28. If L is L3(2) or U3(3), then QL = V x To and <P(QL) = 1 ...

602 3. DETERMINING THE CASES FOR L E .Cj (G, T) on the members of Af normal in T, so that A1 is the only such member. Thus {A, V ...

3.3. NORMALIZERS OF UNIQUENESS GROUPS CONTAIN Na(T) 603 LEMMA 3.3.32. (1) Either QL = V is irreducible, or QL ~ E 32 is the quot ...

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CHAPTER 4 Pushing up in QTKE-groups Recall that in chapter C of Volume I, we proved "local" pushing up theorems in SQTK-groups. ...

606 4. PUSHING UP IN QTKE-GROUPS OM(M+/0 2 (M+)), but just cover M modulo OM(M+/02(M+)) for some M E :E(M+)· For example in the ...