1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

12.7. THE TREATMENT OF A. 6 ON A 6-DIMENSIONAL MODULE 847 Next the preimage Ur is isomorphic to Es and contains V 1 , so by 12.7 ...

848 i2. LARGER GROUPS OVER F2 IN .Cj(G, T) Then H is not solvable by E.1.13, so H is described in E.2.2; in particular (KH n M) ...

12.8. GENERAL TECHNIQUES FOR Ln(2) ON THE NATURAL MODULE 849 I*= Ki~ L2(4). Indeed ID: CD(A)I = 4, so IB: Cs(I)I = IB: Cs(Ak)I:: ...

850 12. LARGER GROUPS OVER F2 IN .Cj (G, T) PROOF. Part (1) is an immediate consequence of Hypothesis 12.8.l. Then (1) implies ( ...

12.8. GENERAL TECHNIQUES FOR Ln(2) ON THE NATURAL MODULE 851 F.8.1 is satisfied for each H E Hz, while the remaining conditions ...

852 i2. LARGER GROUPS OVER F 2 IN .Cj(G, T) with 8i := v = u n U^9 n uh, 8 = (Un U^9 n 8) (Un uh n 8) (U^9 n uh n 8), and 8/V th ...

i2.8. GENERAL TECHNIQUES FOR Ln(2) ON THE NATURAL MODULE 853 (2) 02(I2) = WX, [E,h] =Vi, and 02(I2)/E = W/E tB X/E is the direct ...

854 i2. LARGER GROUPS OVER F2 IN .Cj(G, T) Next [Ziji, W] :::; ZijinU:::; Zu Vi by (1) and symmetry between U and UB. Thus any x ...

12.S. GENERAL TECHNIQUES FOR Ln(2) ON THE NATURAL MODULE S55 Let F := Cu-(X) and suppose F > \/2. Then by (4), Fi. E, while b ...

S56 iz. LARGER GROUPS OVER F2 IN Cf (G, T) 12.8.10.6 that C z9 u (U) = Vf, so that Z[r ~ Zu /Vi ~ Zu, completing the proof of (3 ...

12.g. THE FINAL TREATMENT OF Ln(2), n = 4, 5, ON THE NATURAL MODULE 857 In particular U+ := {O, e5,6, e5,7, e6,7} ::::; 1f 2 ..L ...

858 i2. LARGER GROUPS OVER F2 IN .Cj (G, T) LEMMA 12.9.3. Let 1::::; i < 5 when n = 5,, and i = 1 or 3 when n = 4. Then Li :: ...

12.9. THE FINAL TREATMENT OF Ln(2), n = 4, 5, ON THE NATURAL MODULE S59 M24 in the latter. Now we can repeat our argument in the ...

860 i2. LARGER GROUPS OVER F2 IN .Cj(G, T) Then as Lo is irreducible on D 0 , and 1-=f. b ~ Loi', we conclude b =Do~ E4. Next by ...

i2.9. THE FINAL TREATMENT OF L 0 (2), n = 4, 5, ON THE NATURAL MODULE 86i Suppose first that we are in Case IL Then by the choic ...

862 12. LARGER GROUPS OVER F2 IN .Cj(a, T) establishing (3). D LEMMA 12.9.7. Gi:::; Mv for 1 < i < n. PROOF. Recall Mi ::: ...

i2.9. THE FINAL TREATMENT OF Ln(2), n = 4, 5, ON THE NATURAL MODULE 863 Assume CA(V) =/= l. Conjugating in Na(V9), we may assume ...

(^864) 12. LARGER GROUPS OVER F2 IN Cj(G, T) for u E UH, m(B/CB(u)) S m(V 1 ) = 1, so CB(u) is noncyclic, and hence by 12.9.7, u ...

CHAPTER 13 Mid-size groups over F 2 In this chapter we consider the cases remaining in the Fundamental Setup (3.2.1) after the w ...

866 13. MID-SIZE GROUPS OVER F2 (1) Hypothesis 12.2.3 holds. (2) Ca(v) i M for some v EV#. (3) L/02(L) ~ A5, A6, A6, Ls(2), or G ...