1547845830-Classification_of_Quasithin_Groups_-_Volume_II__Aschbacher_

2.4. THE CASE WHERE rg IS NONEMPTY 547 PROOF. By 2.4.27.1, IT : SI = 2, so (2) holds by 2.3.7.l. By 2.4.27, R = J(S) =To x UoU 0 ...

548 2. CLASSIFYING THE GROUPS WITH JM(T)J = 1 LEMMA 2.4.33. K = E(G2) and F*(G2) = K02(G2). Now suppose that U 2 ::::; Ca(K). Fo ...

2.4. THE CASE WHERE r 0 IS NONEMPTY (2) CR(K) = Ca 2 (K) is cyclic of order 4, and G2/CR(K) S:f Aut(A 6 ). (3) Cr(Lo) = 1 and [T ...

550 2. CLASSIFYING THE GROUPS WITH IM(T)I = 1 Let 02 := G 2 /Ca 2 (K). By 2.4.35, S 1 K 9:! Aut(A5) and hence G2 9:! Aut(A5). In ...

2.5. ELIMINATING THE SHADOWS WITH r 0 EMPTY 551 In addition we define T to consist of the 4-tuples (H, S, T, z) such that HE r 0 ...

552 2. CLASSIFYING THE GROUPS WITH JM(T)I =^1 (2) IfQ1 E VIH(Hu,2), then (U,HuQ 1 ) EU(H). (3) If Lis a component of H which is ...

2.5. ELIMINATING THE SHADOWS WITH r 0 EMPTY 553 We first consider the case where E(H) =/=-1; thus there is a component K of H. A ...

554 2. CLASSIFYING THE GROUPS WITH [M(T)f = i Ko := (K^8 ). Set SK :=Sn K, SK 0 :=Sn Ko, Sc:= Cs(Ko), and fl:= H/Sc. Let x E Nr( ...

2.5. ELIMINATING THE SHADOWS WITH rg EMPTY 555 or by B.2.7 and B.4.2.1, AutA(V) is Sylow in AutALi (V). In the former case V ::: ...

556 2. CLASSIFYING THE GROUPS WITH IM(T)I = 1 is of rank n, and hence [R, w] = [V, u] and [R, w] n Re = [V, u] n Re = 1. Thus [R ...

2.5. ELIMINATING THE SHADOWS WITH rg EMPTY 557 PROOF. First either K appears in case (3) or (4) of 2.5.2, or by 2.5.7, K appears ...

558 2. CLASSIFYING THE GROUPS WITH IM(T)I =^1 PROOF. By Notation 2.5.4, HE r+ so that H = KoS with Ka component of H, Ko = (KH), ...

. 2.5. ELIMINATING THE SHADOWS WITH r~ EMPTY 559 (ii) FI~ Aut(L3(3)) and Cs(Kz) = Sc(y), where y induces an outer automor- phism ...

560 2. CLASSIFYING THE GROUPS WITH IM(T)I = 1 Observe that by 2.5.7 and 2.5.15, we have reduced the list of possibilities for K ...

2.5. ELIMINATING THE SHADOWS WITH r 0 EMPTY (i) K+ = KiK{, Ki -=f. Kl, Ki/021,2(Ki) ~ K, and K = CK+(t)^00 , or (ii) K+ =Ki= [Ki ...

562 2. CLASSIFYING THE GROUPS WITH IM(T)I =^1 Therefore if (i) < 02 (KiCs(i)), then it E t^02 (K;Cs(i)), contradicting it~ t^ ...

2.5. ELIMINATING THE SHADOWS WITH r(; EMPTY 563 (A) If R < Cs(K) choose 8 E Cs(K). (B) If R = Cs(K), choose 8 E Ns(K); we che ...

564 2. CLASSIFYING THE GROUPS WITH IM(T)I = 1 So assume that case (i) holds. Suppose first that L < Lo. We saw that L ~Ki~ L ...

2.5. ELIMINATING THE SHADOWS WITH r~ EMPTY 565 We now eliminate all possibilities for K remaining in 2.5.16 except for the one c ...

566 2. CLASSIFYING THE GROUPS WITH JM(T)J =^1 Set So = Cs(Sc). In any case, (t) = Z(Sc) and SK :::; So, so as IS : SKI :::; 2, I ...